H A N D B O O K
O F
Special Functions Derivatives, Integrals, Series and Other Formulas
H A N D B O O K
O F
Special Functions Derivatives, Integrals, Series and Other Formulas
Yury A. Brychkov Computing Center of the Russian Academy of Sciences Moscow, Russia
Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2008 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-58488-956-4 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
Contents
The Derivatives # " # $!%" & ' & ! ! ( ( )* ' !!" !! '' * *, + + * *, " " & +* *, !" ! & ( # +* *, !" ! ( ' +* *, !" ! . & +* *, !" ! . . ! ! ( +* *, " . !!" !! * *, -- + " " , * * + ! ! / , " * * / + ! ! 0 +* *, " # 1
1.1.
Elementary Functions
1.2.
The Hurwitz Zeta Function ζ (ν, z )
1.3.
The Exponential Integral Ei (z )
1.4.
The Sine si (z ) and Cosine ci (z ) Integrals
1.5.
The Error Functions erf (z ) and erfc (z )
1.6.
The Fresnel Integrals S (z ) and C (z )
1.7.
The Generalized Fresnel Integrals S (z, ν ) and C (z, ν )
1.8.
The Incomplete Gamma Functions γ (ν, z ) and Γ(ν, z )
1.9.
The Parabolic Cylinder Function Dν (z )
!!" !! ** **, .. + + , " 0 ! ! , " * * + +* *, !" !0 !!" !! ** **, + + , " 0 !!" !! * *, + 0 , " * * + !!" !! ** **, ## + + , " 0 !!" !! ** **, '' + + , " 0 !!" !! ** **, && + + , " 0 !!" !! ** **, (( + 0 + , " - +* *, !" ! / +* *, !" ! . +* *, !" ! !!" !! ** **, + + , " " !!" !! ** **, + + , " " !!" ! ** **, + + , " " 1
1.10.
The Bessel Function Jν (z )
1.11.
The Bessel Function Yν (z )
1.12.
The Hankel Functions Hν(1) (z ) and Hν(2) (z )
1.13.
The Modified Bessel Function Iν (z )
1.14.
The Macdonald Function Kν (z )
1.15.
The Struve Functions Hν (z ) and Lν (z )
1.16.
The Anger Jν (z ) and Weber Eν (z ) Functions
1.17.
The Kelvin Functions berν (z ), beiν (z ), kerν (z ) and keiν (z )
1.18.
The Legendre Polynomials Pn (z )
1.19.
The Chebyshev Polynomials Tn (z ) and Un (z )
1.20.
The Hermite Polynomials Hn (z )
1.21.
The Laguerre Polynomials Lλn (z )
1.22.
The Gegenbauer Polynomials Cnλ (z )
1.23.
The Jacobi Polynomials Pn(ρ,σ ) (z )
' ' / / ' & &. # # # #' #' ##( #-#' '& '& '/ '/ & & & &&# &# &# && && && &/
/ & ! ! # +* *, " &./ (. ! ! ' , " * * ' + +* *, !" " (( ( ! ! ' * *, " && + ( ! + * *, " " ( (& ! ! , " * * (( + (& +* *, !" " -. (( - +* *, !" ! -. -. ! ! / -. * *, " / + ! + * *, " " --' !!" ! --&& * *, .. + + * *, " " ( /' /' /' !! 0 0
! ** 0 /' 0"% /' # !! % 0 //&& ' ! ! % !*" & $ "% /( ( ! "% /( - ! "% ! //. ! "% / $%" // " .. ! ! . 0
# . ! ## # . 1
1.24.
The Complete Elliptic Integrals K (z ), E (z ) and D (z )
1.25.
The Legendre Function Pνµ (z )
1.26.
The Kummer Confluent Hypergeometric Function
1 F 1 (a ; b ; z )
1.27.
The Tricomi Confluent Hypergeometric Function Ψ(a; b; z )
1.28.
The Whittaker Functions Mµ,ν (z ) and W µ,ν (z )
1.29.
The Gauss Hypergeometric Function 2 F1 (a, b; c; z )
1.30.
The Generalized Hypergeometric Function p Fq ((ap ); (bq ); z )
Limits
2.1.
Special Functions
H
Indefinite Integrals
3.1.
Elementary Functions
3.2.
Special Functions
L
!! * 00 .' # % .' ./ " ## ! & ### $%" ( ! ! ' '' ##& )* 0 ( ## )) 0 ( / ( 0 ! ! -. ### )) 0* -. -- 0 ## )) 0 - 00 ! ! - ## # )) 0 * 0 - ' # ) " --# 0 0! --'## #### )) 0 " ! -& ! ##### )) 0 0* ( ##' ) ---( 0 0 " ##& ) -/
00 -/ ##'' )) 0 ! ! /. ' / ##'# )) 0 / ' ' 0 / ##'& )) 0 ! ! / / 00! / ##&& )) 0 " / / #& ) 1 H
L
Definite Integrals
4.1.
Elementary Functions
4.2.
The Dilogarithm Li2 (z )
4.3.
The Sine Si (z ) and Cosine ci (z ) Integrals
4.4.
The Error Functions erf (z ), erfi (z ) and erfc (z )
4.5.
The Fresnel Integrals S (z ) and C (z )
4.6.
The Incomplete Gamma Function γ (ν, z )
0 ! ! /' #&#' ) 0 0 /' #& ) #&& ) " /' /& 0 /& #( ) 00 !" //( #( ) ! ! //( #( ) 0 // #(#' ) 0 0 .. * #( ) #(& ) "0 . #(( ) . 0 .## #-- ) .# # ) 0 .' 0 .' #// ) 0 !" .& #/ ) 00 ! ! .#/ ) 0 . #/#' ) 0* #/ ) " 0 #/& ) " # ( ) & 0
. 0 & #. ) # ) " & ( 0 ( # ) 0! / %" # ) 0 / 0 ! ! . # ) 0 ##' ) * # )
0 # ) 0 0 # ) " 0 ## ) 00 ! ! ## ) & ## )
4.7.
The Bessel Function Jν (z )
4.8.
The Bessel Function Yν (z )
4.9.
The Modified Bessel Function Iν (z )
4.10.
The Macdonald Function Kν (z )
4.11.
The Struve Functions Hν (z ) and Lν (z ) H L H H L H L H L
4.12.
The Kelvin Functions berν (z ), beiν (z ), kerν (z ) and keiν (z )
4.13.
The Airy Functions Ai (z ) and Bi (z )
4.14.
The Legendre Polynomials Pn (z )
0 0 #####' )) " # 0 ' 0 ##' )) # #' ) 0" #& 00 # ##&& )) # # 0
##&&# )) "0 #& #& 00! #( ##(( )) -/ " #( # 0 0 '# ##((# )) 0
' ##((& )) "0 ' ' '# 0 '# #-- ) 0 '' #- ) 0 '& #- ) ## ) "0 '& '( 0 '( #// ) 0 '( #/ ) # ) "0 & & ! !0 & #.. ) !0 & #. ) 0 ! & #. ) # # ) "0 &#' !0 ! 0 &&' ## )) " %" &( 0 ! ! ### )) 0* &( #' ) 0 (#
4.15.
The Chebyshev Polynomials Tn (z )
4.16.
The Chebyshev Polynomials Un (z )
4.17.
The Hermite Polynomials Hn (z )
4.18.
The Laguerre Polynomials Lλn (z )
4.19.
The Gegenbauer Polynomials Cnλ (z )
4.20.
The Jacobi Polynomials Pn(ρ,σ ) (z )
4.21.
The Complete Elliptic Integral K (z ) K K K K K
0 (# #& ) 0 (' #-( ) 0 (' #/ ) 0 (' # . ) 0 (& #) 0 (( #) 0 0 (# ) # ) " (-/ 0 ((/ # ) !" !%" 0 # ) 0 ! ! - & # ) 0* ##' ) 0 -/ # ) 0 / #& ) 0 / #-( ) 0 / #/ ) 0 / # . ) 0 / # # #) 0 / & #) 0 / / # ) "0 0 /& //& #) "0 # #) .. 0 % .. # ) . # ) "0 0 . 0 . ## ) 0 .# ## ) 0 ! ! .& ## ) 0 .& ###' ) # # ) "0 .& ./ ./ ' ./ ' " 0 ' K K K K K K
H
L
K
K
4.22.
The Complete Elliptic Integral E (z ) E E E E E E E E E E E
H
L
E
E E
4.23.
4.24.
K
The Complete Elliptic Integral D (z ) D D K
The Generalized Hypergeometric Function p Fq ((ap ); (bq ); z )
K
Finite Sums
5.1.
E
The Psi Function ψ (z )
E
'' 0 " ' # ' 0 # ' " # ' ''# 0 0 '' "0 '## " ( ''''
- 0
" '' "0 / / '& 0 / . ''( 0 . ' 0 " '(( " ' '(# & ''(' 0 (& " / ''-- 0 / "
''-- 0 '-#' ' " '-& ' ( ''// 0 0 " #.( '/ # ''//#' " ## '/& "0 ## ## '. ##
5.2.
The Incomplete Gamma Functions γ (ν, z ) and Γ(ν, z )
5.3.
The Bessel Function Jν (z )
5.4.
The Modified Bessel Function Iν (z )
5.5.
The Macdonald Function Kν (z )
5.6.
The Struve Functions Hν (z ) and Lν (z ) H L
5.7.
The Legendre Polynomials Pn (z )
5.8.
The Chebyshev Polynomials Tn (z ) and Un (z )
5.9.
The Hermite Polynomials Hn (z )
5.10.
The Laguerre Polynomials Lλn (z )
#& '. #'. 0" '' '.# '( '.' "0 '/ '.& 0" & '.( & '.- "0 & & ' ' &' ' &' 0" ( '# '' "0 ( - '& '( 0" -'- "0 /. / ' !! / ' / ' !0" #. '# "0 ! #.' '' ! #.' '& ! 0" #.'( "0 ! #. ' # # # ''# # 0 0" '## # " # # '''' # 0 " ## # '& # ##
5.11.
The Gegenbauer Polynomials Cnλ (z )
5.12.
The Jacobi Polynomials Pn(ρ,σ ) (z )
5.13.
The Legendre Function Pνµ (z )
5.14.
The Kummer Confluent Hypergeometric Function
1 F 1 (a ; b ; z )
5.15.
The Tricomi Confluent Hypergeometric Function Ψ(a; b; z )
5.16.
The Gauss Hypergeometric Function 2 F1 (a, b; c; z )
'& 00" ' '& " # #('(
0 # # # '( '(
" # 0 '(# " ## '(' "0 ' '(& ##' #& ''-- # & ! " % #( #/ / # / && # ! / " # ! . # &&# %" # # & 0 # 0 && ##( ##( " &# ##'/ # & #''
# ##'' &# #0 #' &# #' &&### "0# # #'
#
#00 ##' &&'' #'' # #
0
' # # # #'&' 0 & # "
#0 # #'/ #'/
&& "0 #'/. && #& 1
5.17.
The Generalized Hypergeometric Function p Fq ((ap ); (bq ); z )
5.18.
Multiple Sums
Infinite Series
6.1.
Elementary Functions
6.2.
The Psi Function ψ (z )
6.3.
The Hurwitz Zeta Function ζ (s, z )
6.4.
The Sine Si (z ) and Cosine ci (z ) Integrals
6.5.
The Fresnel Integrals S (x) and C (x)
6.6.
The Incomplete Gamma Function γ (ν, z )
. #& 0 &( % #&. #& &-- #& , &- ! #&' &- #& &-#' 0 #&& &- # #&( &-& #0 #&( &--( "0 # #(.. &-/ "0 #0 #( &- . #0 # #( &- # #0 # #( &- #( &- #( #(# &- & # #('' #(' / &/ 0 #( &/ "0 #(& &/ #(( -. 0 &/#' " # & #-. - # 0 #- &.. # #- &.
0 #- &. ## &.#' # 0 # #0 # #--# &. #0 # #-'# &.& "0 # #-' # &( -& # 0 #--& & #& & 0 " & #- #-&# -/ # 0 #-/ & #-/ & 1
6.7.
The Parabolic Cylinder Function Dν (z )
6.8.
The Bessel Functions Jν (z ) and Yν (z )
6.9.
The Modified Bessel Function Iν (z )
6.10.
The Struve Functions Hν (z ) and Lν (z ) H L H H H H H H
6.11.
The Legendre Polynomials Pn (z )
6.12.
The Chebyshev Polynomials Tk (z ) and Uk (z )
0 #/. & #/. &# / # 0 / && # 0 " #/
0 #/ & #//# " & #' 0
" ## & /' # #/' &# #/& 0" &# "0 #/( &# "0 # #/( &##' #/( &# 0" #/ &#& 0 &#( " #/// # #// &'' '.. 0" &' 0 " '. &' '. &'#' & 0" '.# '.' ! && ! 0 '.' '.& " && 0 && " ! ! '. ( '. &&#' ! 0 '.( && " ( '.
'.&&(( ' 0
0 ' " &&((# "0 '#( # '( &&((&' '
0" ' &(-( " 0 ' & &( 1
6.13.
Hermite Polynomials Hn (z )
6.14.
The Laguerre Polynomials Lλn (z )
6.15.
The Gegenbauer Polynomials Cnλ (z )
6.16.
The Jacobi Polynomials Pn(ρ,σ ) (z )
6.17.
The Generalized Hypergeometric Function p Fq ((ap ); (bq ); z )
' ( ' ' ( ( ' ( ' ( ! " (( !" 0 ' -( %0 ! ! ' ( !!" 0 ' / 0 ((#' ! '# * ! 0 ''# ' ((&( ! % 0 # !
* ((/ ! 0"% 0 ''#& / # ! ! ! 0 % " *" '''#/ % ((. !$ % % '' " ( !! "% ! '' ( ! ''' "% ((#' !"% ! % '' ! " " ''/ ((&( ! " " '&. ! 0 (- '& ''& '& -- !!!!%" & '-/ - !!%" &. --#' !!!!%" & %" &-/ --& !!!!%" %" & & --( !!%" & %"
!%" & --/. !! ! %" & -- !!!!%" &' &/ - !!%" &/ %" --#' !!!!%"
& & %" -& !!%" & 1 The Connection Formulas
7.1.
Elementary Functions
7.2.
Special Functions
H
J
L
E
K E
Representations of Hypergeometric Functions and of the Meijer G Function
8.1.
The Hypergeometric Functions
---( !!!!%" & & %" --/. !!!!%" & & '# -- !!!!%" & %" & & %" - ! 0! %" & ( & & -- & / ' -- % &&' " &'' &&/ 0 &( / ))00 % &( (a p ) (b q )
8.2.
The Meijer Function Gm,n z p,q
8.3.
Representation in Terms of Hypergeometric Functions
1
Preface
$ !% &'(!#%)!"* +" ,& ")-.!% & !"-) /" * !"# ! ,& )!-$,.))!%< )-= &"45% / $6(!# $%(#)!"%" "4%#!&' 710;23 ) + 0 8
7
3 9 ; 78 : 1023 > .!% &"
@
A 7 ! $
5 " ?
8
8
B
>
9 C
7
"
7
9
7
EMENNN B
F K L
G H I J DE E E E E E
Chapter 1
The Derivatives 1.1. Elementary Functions 1.1.1. General formulas 1.
2.
3. 4. 5.
6.
! '#" " . ')(* $ ! % ,+ &-
¤ & $ %
'65 $ (87! % . & -21 0/ 0 43 ¢ $9% ')( $ (:7! % £ / -;&- <+ &- / 7 "" ' . 7 & -21 ' ¢ ¢ I G H ) ' ( : ( 7 ! $ % ,+ &- F G ED EF 3 3 KJ $L 7 7 EDM &-21 EF GIH 3 ¢ -;&-21ON F G < PQ , & N & ¤ 1 & R 0& ¤¤4¤ ST0& U X[<\]W ')( $ Y &- -< Z Z Z [<\#U ^ 7 EDM SEa F G S D S -21Ab
2. 3. 4.
= '8>?7A@CB
U4. & - U ')( $ XW4. ' + & + 0& ¤¤4¤ $ $ L V V L
U4 J 1 XW4 J (_BB B<( $ ( " [<\#U . U X[<\#U4. ¤ $( " & ` S + 2- 1 & R S + &- -
1.1.2. Algebraic functions 1.
= '8>?7A@CB
&c# 43 ¢ 43Pd2 & c -; ¤ . &e< fg3h& i f & e ;- 0fg3h& 0i -;j +ke -;9l i -; F . &e< fg3h& i &e0fg3h& i -; j + - e - i -21Rl i -; F ¢ . m &e< fg3h& i &e -; 0fg3h& i j +ke -;9lC- e - i -21 F m
¢ 3 O G ¤ m m ¤ 3 :G 5n ¤ (* G
5.
6. 7.
8.
( &! ' m c f & c F G c -; 0f 1 ( L' J
3 ¢ 0 fI& c -; - c F O 5 m 5 m -21 0 f & -21 m 5 £ 3 ¢ a 1 f -21 f & -;&-21 ¢ 3 F G - c -21 ] fg3h& c £ -
9.
c
10.
11.
12.
13.
14.
15.
16.
17.
18.
,
& c -; c -; a 1 F ¢ m G ¤ ¤ G
. 5n O + a 1 a F m 5 G! ¤ m 1
'! % m ¤ a (1 L ' - c -21 0fg3h& c -; c -; 1 F#" G J
5 m fg3h& &- c -21 4 3 ¢ f
c -; 0fg3h& - c -21$ - c F m G ¤ &- c -21 0f 7 & c ( 9! ' m ¤ F3 G ¢ a c ; 8G f & ( L ' &- c -21 c -; 1 F 1 % J
- c -!&4 0fg3h& c &L ' m ¤ 3 ' % a ( 6J ( 1 L '%& f -21 O - c -21 fg3h& c -; c -;a 1 1 F#" G ' J
1L'
( f -21 0fg3h& c -; c -; a 1 F " m G ¤ &-21 0fg3h& c J 1 J L' a 1 0f & c '65_7! % 3 £ - &-21 ( 5_ 7X! '%& $ fI c a 1 - c a -;&-21 5 m-, ¤ 1 )+* '! % &-21 0f & c £ - ( 5 7! ' -21 f- & c. - c -; )+* m 5 , ¤ m 5 . 0f & -21 f 0f & - + a 1 j F m 5 m G ¤ f & &-21 f f & -21 j F " ¢ m G ¤ 0 f & a 1 f a 1 j a 1 F " ¢ m G ¤ 0 m f & -;&-21 0 f & -;&-21 j ) * m 5 , ¤ 0 ,
19. 20. 21. 22. 23.
24.
25. 26. 27. 28. 29. 30. 31. 32. 33.
34. 35.
0f & -;&-!&4 f -21 ] f- & -;&-21Oj 0 f & f j F ¢ m G ¤ -;&-21 0 f & 3 ¢ -;&-21 j F ¢
m 5 , ¤ a m
1 )+* m ¤ G
5m : , ¤ &-21 f h& -;&-21 43 ¢ -21 0f h & -;&-21 j )+* &-21 0 f & 3PfI -21 j F " m G ¤ &L ' a 1 0 f & a 1 'gJ 5 7 f 1 0 f & 1 F ¢ m G ¤ L' # J 'Q& 5_ 7! f a 1 1 a 1 F " ¢ m G ¤ 7 &-21 0f & a 1 F G f a 1 -21 a 1 F#" ¢ m G ¤ 7 a 1 0f & &-21 F G f 1 0f & -21 F " ¢ m G ¤ 7 ¢ a 1 0fg3h& &-21 43 F G f a 1 fg3h& -21 a 1 F#" m G ¤ 7 &-21 0 f & &-21 F G f -21 f & -21 #F " ¢ m G ¤ 7 m &-21 0 f & -; F G -21 f & -; )+* m 5 , ¤ 7 m a 1 0 f & -;&-21 F G 1 f & -;&-21 )+* m 5n , ¤ 43 ¢ F 7 G f & -;&-21 a 1 ) * m 5 , ¤ &-21 0 f & -;&-21 m F 7 G fI& -21 0 f & -;&-21 a 1 ) * m 5 , ¤ 43 ¢ F 7 G -21 0 f h -;&-21 )+* m 5: , ¤ a 1 0 f & -;&- ( m ¤ 43 ¢ £ - O'6'65 5_77X! ! % % 1
; & f & F 5 m G 3
36. 37. 38.
39. 40. 41.
42. 43. 44. 45. 46. 47. 48.
49.
L ' 1
m # J 'Q& 5_ 7! F m G f & -;&-!&4 a 1 5n , ¤ m +) * ( ' 9 ! ' 0f9 3h
c 3 £ fI ' ( &! ' 0f9 3h c -; c -; a 1 F m G ¤ 0f9 3h
-21 ( m a . f -21 fg3h& -;&-21 ¢ 3 F E 5 m G + 1 a 1 F ( m G ¤ m
0f9 3h
-21 £ f -21 f9 3h O -;&-21 a 1 F m (8 G ¤ m
a 1 0f9 3h
-21 £ ¢ f -21 ; f9 3h O -;&-!&4 a 1 F m (8 G ¤ -21 0f & -21 43 ¢ - + a 1 . 0f & - + a 1 . j F 5 m m 5 . 0f9
-21 R 43 ¢ f9 - + a 1 j F m 5 G ¤ 0f9 3h
43 £ fI j F m G ¤ 7 0f9 3h
&-21 R 43 £ fI F G f9 3h
-21 F m G ¤ 7 0fIQ3h
&-21 R 43fI F G 0fIQ3h
-21 F" m G ¤ ( m ! ' a 0f 3h 1 '65_7 F G 0f 3h 1 F m G ¤ m £ -;&-21 0f9 3h
43 fI -;&-21 j F G ¤ &- c -21 0 3hf9
c ('9! ' £ fI ')( &! &- c -21 0 3hf c -; c -; a 1 F ' 7 5 m F $ f G 1
3 ¢ F 7 G 0 f9
- + a 1 . F O'6 5_7 m
1.1.3. The exponential function 1. 2.
3
k c - c -; - c -; 0fI& ¤ m c -R 43 ¢ c -; -R - c -21 F G ¤ 5
G ¤
m ¤ G ¤ m G
W W 3A f R/ f& ¤ 3. ED H W W m f
-;&-21 R F G ¤ 4. EDM &-21 R H W W ¢ H 5. ED 3 f - / f& ¤ - W W m 6. EDM &-21 #-R H f -;&-21 #-R F G ¤ m a . 7. / 8F G 1 + 1A- &-21 f;/ 1 -; 0f;/ ¤ 7 m a . 8. D H / *F G 1 - + 1 a a 1 f;/ -;&-21 f;/ 0 ¤ 9. k &-21 R m m ¤ 3 ¢ / *F m G a 1 - + a . D G F
F &-21
1 -; GIH 10. k &-21 R m 43 ¢ / F m G a 1 - + & a . D a F m G ¤ F 1
-;&-21 GIH m']" ' & . ¤ ¢ 3 11. - I+ 1A- &-21 f / 7 m'#" '% & ¤ ¢ a . 43 12. D - H - + 1 a 1 0f / m ¤ ']m " '% & a . 13. &-21 -R - + &-21 F G m ¤ ']m " '% & a . 14. k -;&-21 #-R - + & a 1 F G m a . U . . 15. D + -21 3 ¢ a 1 R/ *F G 1 + 1A- + f;/ H 1 -; = 7 @CB m a 1
. . ¢ G +
16. 43 / 8F 1A- + 0f;/ = 7] X@CB & 2 1 7 . U 17. D + -21 H 3 ¢ a a 1 R/ *F m G a 1 - + a 1 . + . 0f;/ = 7 @CB a 1
/ F m G a 1 - + a 1 . + . 0f / = 7] X@CB 18. ; -;&-21
5
1.1.4. Hyperbolic functions 1.
&c
2.
c
' % fI& 0 &c -; ' % fI& 0 c -;
c ;- 4 3fI& <3 #- c -; 4 3fI& -
¤ c ;- fI& ¤ c -; fI&
fI& 0 ")" $ 5 7! " £ 3fI ¢ S $ L ¤ " 5_7! 3 "'& 8 4 3 J S W W ( ! ' ( 7! ' f R/ f & 3 f - 0/ f& ¤ 4. 0fI
W W ( ! ' ( 7X! ' f R/ f& f V- 0/ f& ¤ 5. fI 0 / 8F m G a 1 I+ 1A- . ¤ 6. 0f;/ 0 1 -; f;/ / 8F m G a 1 I+ 1A- . ¤ 7. f;/ 0 &-21 f;/ ! m m a . H / 8F G 1 - + 1 a a 1 f;/ ¤ 8. D ! m m a . H / *F G 1 - + 1 a -;&-21 0f / ¤ 9. ED a ¢ 10. / 3 ?F 5 G 1 7 ( &B B B ( 5 &B BB 5 #$ & 1 1 1 1 ¤ Y a N a
& 1 & ( &B BB & ( & 5 &B BB & 5 "! ( 7
'
% ' ] ( X 7 ! ' % % ¢ H 43 a 1 '%& '%& 11. ED #$ 7V( &B B B V( &B BB ¤ Y a N a
& 7 ( 9BB B #7( #7,5 &B B B #7<5 ! ( 7 & a H £ 43 ¢ F 5 G 1 12. D 7 ( &B BB ( 5 9BB B 5 #$ 1 1 1 1 ¤ Y a N a
& & ( 9BB B & ( & 5 &B B B & 5 "! 7
3.
:
' % ED H 4 3 ¢ a 1 '%& #$ 7 ( & B B B ( &B B B # ( 7! ' ' % ' & a & N a 7 ( 9 BB B #7( # 7,5 &B B B #7<5 7 ¤ ! m m ¤ 3 ¢ / F m G a 1 - + a . 14. D &-21 F GIH F 1 -; G m m ¤ 3 ¢ / *F m G a 1 - + a . 15. ED &-21 F GIH F &-21 G m 3 ¢ / F m G a 1 - + & a . a F m G ¤ 16. D &-21 F GIH 1 m m ¤ 3 ¢ / F m G a 1 - + & a . 17. ED &-21 F GIH F -;&-21 G 18. 0f / '%& f a 1
I+ 1A- . & -21 $&% '6 ')5 ( $ $ ! ! 43fI - - a 0f / -; 1
= '*>?7R@CB 13.
0 f / ' & f a 1
I+ 1A- . & -21 $9% '6 ')5 ( $ $ ! ! 43fI - - & - -21 f / = '*>?7R@CB m ! 20. ' & f a 1 - + a 1 . & -21 $9% '6 ')5 ( $ $ ! ! 3PfI - - a f / &- 1
= '*>?7R@CB m ! 21. '%& f a 1 - + a 1 . & -21 $&% '6 ' 5 ( $ $ ! ! 43fI - - -;&-21 0f / = '*>?7R@CB 19.
1.1.5. Trigonometric functions 1.
0fI& f
F fI
' ¤ G <
2. 3.
4.
5.
6.
7.
8.
9.
0 fI& f
F fI
;
' ¤ G
. 0fI& 4 3 ¢ + a 1 f $ £ Y S J L 3 " 3 ¢ + a 1 . f ( 7X" ! 1 (* . S $ +
¢ Y
5 L 3 J
' 5_7 ( 7 ! " 6 " J $ 5 7 L 1 7 ( D 3 £ fIQ3
a
1 fI& ( 7! ' H
= = @ C! @CB
'65_7 $ £ 5_7 L SEa 1 fI& S J L 3 J$ ' a
! = @ O !#B
fI&
. 7 " + - ¢ S fI& S 43 J Y SEa 0fI& 43 ¢ $ L £L ¢ J " $ 5_7X ! £ AfI ¢ S 5 7! 3 "'& 8 S 3
$
3 ¢ + a 1 . f
5 L
'
= @ ! #B !
$ J L
= = @ !C@CB
' 3 £ a 1 m 5 ' % ! %' & m' m m . 5n . 5n m ! '%& D + F G 3 + F 9G H ( m 5 m ¤ m ' '%& . * . n ¢ 3 43 ! D + F G 3 + F 3 IG H
fI& 0 " 5_7 3 ¢ f ( 7" ! '6 5_7 L J $ S . S 1 $ (* Y + - - 3 ¢ 5 L a J 3 ¢ f fI& 7 " 43 S 1 . $ +
a ¢ Y P5 L
3 J L
J
$
0fI&
L 3 £
SEa 1 fI&
'
!
= @ C7X O ! B
¢ S $ L £ ¢ J ' 0fI& ! = @ C7 O ! B
m ' %' & . 5 m . m 10. D + F G 3 + F G9H ! m '' & . ( m . m ¢ G 3 + F3 G9H ¤ 3 43 ! D + F $"'& " m ]! ¢ £ a 43 1 fI 11. 0fI& 7( ( 7! 1 ' $ ¢ Y D 3 4fI$ H 3 ¢ S J L = = @ C7 !C@CB S 1 $ 7" ¢ £ a 43 4 1 fI fI& 3 ¢ S J L 12. 1 S 1 a . 8 ( 7 $ ' @ ! B Y + V- 3 ¢ Q( Q5_7 L - a 1 fI& = ! J 43 ¢ a 1 ' ' % & m
m ' '&
m ' '&
. 7 m . 7 m D + F G 3 3 ¢ + F 3 IG H ¤ $ 7" . £ ¢ 14. fI& 0 43 + &-21 C fI fI& 3 ¢ S J L S 1 a . ' (:7 Y + V- 43 ¢ Q$ ( 5_7 L - a 1 fI& ! = @ ! B J
13.
15.
16.
c
17.
c
. m . D 3 ¢ + F]3 G 3 + ' % fI& 0 c -; c -; 43 ' % 0fI& c -; c -; 43
' fI
0 ( 7X! f + &- . D W & 19. 0fI'
( 7X! f D W & m 20. ED &-21 H 3PfI -;&-21 F
18.
21.
D &-21
m H 3PfI -;&-21
P
/ fE3
m
m F
m F ¢ G9H ¤ AfI& 3 - AfI& -
/ f ' ¤ G ' ¤ G
c -; AfI& ¤
c -; AfI& ¤
-
W
W
/ f H ¤
/ f H ¤
;
m a . f / 0 / *F G 1 I + 1A- 1 -; f / ¤ 3 ¢ / *F m G a 1 I+ 1A- . ¤ 23. 0f;/ &-21 0f;/ S 24. 0f;/ < ' & S - + 1 a + -21 . [ .
m 43 ¢ S ( &(']" '#" ¢ $ L 3 3 8 J 7 ( ( 7X! ( £) a ) £ Y ( 3 1 1 -; 4 3 4f;/ , 7 h 5 ( 7! £! a > 7R@CB ¢ 43 3 1 &-21 3 £! 4f;/ = ? S 25. 0f;/ ' & S £) a ¤ £) 43 ¢ ( & '% & m '#" 3 8 J $ L 3 1 &-21 4 3 4 f / 22.
26. 27. 28.
29.
30.
31.
m 2! m a . H 3 ¢ / *F G 1 - + 1 a a 1 0f / ¤ ! m m a . ED H / *F G 1 - + 1 a -;&-21 f;/ ¤ a / ?F ( G 1 7 ( &B BB ( 5 9BB B 5 #$ & 1 1 1 1 Y a N a
& 1 & ( &B BB & ( & 5 &B B B & 5 ! ( 7
' %
% ] ( 7X ! ' ' %
¢ a D H 3 1 ' & '%& 7 V( &B BB V( &B B B ( 7 #$ ! Y a N a
& 7 ( &B BB #7( #7<5 &B B B #7<5 a £ ED H #F ( G 1 7 ( 9BB B ( 5 &B B B 5 #$ 1 1 1 1 Y a N a
( &
B
B B ( 5 &
B B B 5 7
& & & & & !
£ ¢ D H 43 -;&-21 7 V( 9BB B V( &B B B 7 #$ ! ¢ '%' & % Y a N a 3 4 3
& 7 ( &B BB #7( #7<5 &B B B V7<5 ED
N
¤
¤
¤
¤
32.
ED &-21
33.
ED &-21
34.
D &-21
35.
ED &-21
m ¢ m a m ¤ . H 4 3 / 8F G 1 - + a 1 -; F G
m m a m ¤ . H / 8F G 1 - + a & a 1 F G
m m a m ¤ a . H / F G 1 - + &-21 F G
m ¢ m a m ¤ . H 4 3 / 8F G 1 - + a & -;&-21 F G
. f / 0 £ - &-21 O/ f a 1
I+ 1A- '65 ! Y & -21 43fI - $9% ')( $ $ ! - a -; 1
. 0f / 3 ¢ £ - &-21 / f a 1
I+ 1A- '65 ! Y & -21 f - $9% ')( $ $ ! - &- -21
m ! 43 ¢ £ - &-21 / f a 1 O - + 1 a . '65 ! Y & -21 f - $9% ')( $ $ ! - a &- 1
m ! £ - &-21 / f a 1 - + 1 a . '65 ! Y & -21 43 ¢ f - $&% ')( $ $ ! - -;&-21
36.
f / = '8>?7A@ B
37.
f / = '8>?7A@CB
38.
f / = '8>?7A@CB
39.
0f / = '8>?7A@ B
H a 1;D H a D H a & D H D H a 1 D H
41.
42.
43.
44.
45.
RED
40.
R R R R R
3#
3#
643 ¢ £ a 1
3V
3 ¢ a 1 £ a 1
3V
¤
¤
¤
& ¤
& ¤
& ¤
R a & D
47.
R a D
46.
48.
49.
50.
51.
H 643 ¢ £ a 1
:
H 3 ¢ a 1 £ a 1
¤
a 1
& ¤
m . 0f;/ ¢ / 8 G F I +
A 1 f;/ Q ' 5 ! £ £ <1 f / ' <1 f / H ¢ a 1 / *F m G a 1 I+ 1A- . 0f;/ f;/ O'65 ! f;/ £ O'65 ! £ D f / K3 f / H 2 1 2 1 7 ¢ / *F m G a 1 - + a 1 . 0f;/ f;/ f;/ O'65 ! Q ' 5 ! £ ' / D
f f / £ H 2 <1
2<1
7 f;/ 0f / f;/ ¢ a 1 / *F m G a 1 - + a 1 . O'65 ! 'Q5 ! £ £ D f / E3 f / H 1 1
f;/ O'65 ! Y D Y Y Y
¤
¤
¤
¤
1.1.6. The logarithmic function 1.
2.
/ / f 7X! % 5 m 43 ¢ & -21 ')(: -; f9 -; j & -21 5 m , ) k &-21 / f / f 5n m ') O(8 7! % 3 ') O(8 7! % 5 m j F m m 5 m G &-21
3.
D F $
4.
D &-21
5.
f G9H 43 ¢ & -21 3 ¢ 0 9f O ;- j F &-21 5 m G
= '*>?7R@CB = '*>?7R@CB
F f f GIH ) ' (87! % ) ' (:7X! % m f f9
-; j & -21 F 5 m G 3 m m 5
ED m (8
-;&-21 &2- 1 F m * ( G
= '*>?7R@CB
!,
= '*>?7R@CB = '8>?7A@CB
m 5 m (
13. ED 1 PED -21 0f 3h& &-21 m ( HOH 43 ¢ F 7 G f f 3h& -;&-21 m 5 14. ED &-21 PED f 3h& &-21 m ( HOH 43 ¢ F 7 G f -21 f 3h& -;&-21 ¢ ¢ ¡ 15. &-21 3hfI& 3hfI& 0 ¢ ¢ ¡ 16. 3hfI& &-21 3hfI& 0 ¢ ¢ a ¢ 17. 1 3hfI -21 -21 3hfI 3hfI& 43 ¢ f ¢ 3hfI& -;&-21 ¢ 4
f ¢ 3hfI& ¤ a ¢ 18. 1 3hfI -21 3hfI& 0 6.
a 1;D m 5 D m (
1.1.7. Inverse trigonometric functions 1.
= '*>?7R@CB = '*>?7R@CB = '8>?7A@CB = '8>?7A@CB m 5 ¤ m (
m 5 ¤ m ( m 5 ¤ m ( = '8>?7A@CB = '8>?7A@CB ¢ 3hfI& ¤
m fI& 0 3A &-21 3 ¢ f ¢ h 3 f9
-; j &-21 F 7( m G ' >?7R@CB =* 3
2.
3.
4.
5.
6.
7.
0 fI& m 3 ¢ &-21 3 ¢ f ¢ h 3 f9
-; j & -21 F 7 ( m
fI& 0 7 43 ¢ £ 3 ¢ f9 a 1 ; ¢ f9 O
-;&-21
&-21 F 7<5 m
a 1 fI& 0 43 ¢ £ f9 a 1 ¢ f9 O
-;&-21 a F
1
fI& 0 7 43 ¢ a 1 £ 3 ¢ f9 a 1 ; ¢ f9 O
-;&-21 F
&-21 7<5 m
a 1 fI& 0 43 ¢ a 1 £ f9 a 1 ¢ f9 O
-;&-21 a F
1 ( f;! / ']" 0 7( m ¢ 3 f 0Q3hf -; j &-21 m m (8 )
<
G
= '*>?7R@CB G
= '*>?7R@CB 7
¤ 7<5 m G
= '*>h7A@CB G 7
¤ 7<5 m G ,
f;/ ') 0(: 7X! % . fI 1 -; f $ ¢ -; j + 1 -;9lC-; £ f ¢ &-21 m 9. EDM &-21 H '#" ( m 3 3 ¢ f -21 0Q3hf9
-; j &-21 ) m m (8 , m 10. D &-21 H ' ( 7! 3 ¢ fI &-!&4 0 f -; j + 1 -;9lC-; . m ¢ , ) &-21 ¢ ¢ a 11. k &-21 3hf & 1 M 3hf & -21 f;/ F 7 G f9 -21 ¢ a 12. k 1 3hf9
& &-21 k -21 0f;/ F 27 G f9 ¢ 3hf9 O& -21 ¢ ¢ a 13. k -21 3hf9
& 1 k &-21 3hf9
& -21 f / 43V -; £ -;&-21 8.
5
= '*>?7R@CB = '*>?7R@CB = '*>?7R@CB = '*>?7R@CB f;/ ¤ 0f / ¤ 0f;/ ¤
14.
15.
16. 17. 18. 19. 20. 21.
22.
23.
24.
k a 1 ¢ h 3 f & a 1 k -21 ¢ h 3 f & -21 f;/ f9 f / ¤ a 1 ¢ 3hf9
& -21 -21 ¢ 3hf9
& &-21 f / £ 3 m ¢ 3hf9
& -;&-21 f;/ ¤ , ) k 1 ¢ 3hf9
& &-21 &-21 f;/ 0 ¡ = '8>?7A@CB
k &-21 ¢ f9
& P f;/ ¡ k -21 ¢ f9
& Pk &-21 f / £ F3 k a 1 ¢ f9
& Pk -21 f;/ £ k &-21 M ¢ f & &-21 f;/ F 7 G f -21
k a 1 k -21 ¢ f9
& &-21 F 7 G f9
7 G -;&-21
0f;/ ¤
f / ¤
¢ f & -;&-21
f;/ ¤
0f;/ ¢ 9f
& -;&-21
f;/ ¤
3 m , )
1.2.1. Derivatives with respect to the argument
2.
fI& EDM &-21 F
3PfI ; < If & ¤ m GIH f -;& -21 F < m G ¤
1.2.2. Derivatives with respect to the parameter 1.
&
M ]! ¤
!;
= '8>?7A@CB = '8>?7A@CB
1.2. The Hurwitz Zeta Function ζ(ν, z) 1.
= '8>?7A@CB = '8>?7A@CB
k &-21 ¢ h 3 f9
& 1 ¢ h 3 f9 O& &-21 0f / ¡ k &-21 ¢ 3hf & &-21 f;/ ¡ k 1 ¢ f9
& &-21 f;/ ¡
2.
3. 4.
5.
6.
7 ' a $ £) 3 £ F '*G 7 - 1 3 $ ' " £! 3 £ ( 7X! " & -21 . ( 7! " ] $ (:7X! % " 3 ' ! ' + -21 F 'PG 3 O' ! " 1 7 Y & -21 ' F £) ' G ' " 43 £! ¢ = ' 7 1
F G a 3 ' ' £ 1A- 3 ¢ 3 £ ¢ - 1 C7( " ' ! 7 F G a C7( " ' !0' - 1 ( 7X! ' 7 ] ' " ' # ' " '#" + &-21 . F G 3 ' " ' (:7 3 £ ¢ C7 (* " ' ! 7 G a ' F
C7( - " ' ! 1 ( 7! " . 7 £ % ' & # ' " # ' " 3 ' " ' + &-21 F G 7( 3 '#" 3 £ ¢ C7( " ' !A C7<5n " ' ! F G a '
- 1 C7 ( " ' ! £ ' & ' " ' C7(* ! ( 7X! " "" 5_7X! . 7 # ' " ] ' " # ' " ] ' "
+
F G
& 2 1 '
7 (* " ' 7( " ' 3 £ ¢
31
! =
@ O!C@CB
= = @ # C7 !C@MB
= = @ C7 !C@CB
= = @ C7 O!C@CB
= = @& !C@MB
1.3. The Exponential Integral Ei (z) 1.3.1. Derivatives with respect to the argument 1. 2.
" & 2- 1 m ]! ¢ ¢ 3PfI& 3 &-21 3 -; - &$ % 3 ¢ -; - -; fI& &-21 :
= '8>?7A@ B = '8>?7A@ B
$! "!$ #
" & -21 m ]! ¢ GH Q 3 3 -21 - R $&% 3. EDM &-21 F 3 9 3 ¢ 3 ¢ -21 -R -; F m G 4. &-21 " & -21 ( 7X! "'& $&% f 43fI& f m ]! 5. 3fI& 0 m 6. EDM &-21 R F]3 GIH 43fI -;&-21 R F]3 m G 43 ¢ f &-21 -; & -21 3 ¢ F G m 3 ¢
-;&-21 43fI& ¤ 7. -21 - P 3PfI& a - R ED -21 R F3 m G9H H 43 ¢ 4 F]3 m G ¤ 8. EDM 1 f 3PfI& ¤ 9. V- 3PfI& m m 10. ED - R PED &-21 R F]3 GIHOH f -;&-21 F]3 G ¤
m
= '8>?7A@ B = '8>?7A@ B = '8>?7A@ B = '*>h7A@ B
1.4. The Sine si (z) and Cosine ci (z) Integrals 1.4.1. Derivatives with respect to the argument 1. 2.
3. 4.
5.
')(:7X! % -; 0fI& m EDM &-21 F GIH 7! % 43 ¢ ' (8 D ')(:7X! % -; 0fI&
&-;-2 1 3AfI& 3 V -
m G 3 #- R R -; &-21 F 3 -; 43 AfI& - &-21
&-;-2 1 AfI&
= '8>?7A@ B
m ;&- -2 1 F 9G H &-;-2 1 AfI&
= '*>?7R@CB = '8>?7A@ B
m EDM &-21 F G9H 4 3 ¢ ') O(8 7! % D R ;- F3 m G #- R ;- F m G9H &2- 1 &2- 1 & <3 0& 3 ¢ FO63 ' G & 3 F Q3 ' G &
7
= '*>?7R@CB
' 7 ' & 3 )£ ¢ 3P
¤
1 !<
6.
&
0& 4 3 ¢
F Q3
' G
F Q3
;
' G
0&
7 0& 3 '%& 3 !£ 3 ¤ 1
1.5. The Error Functions erf (z) and erfc (z) 1.5.1. Derivatives with respect to the argument 1. 2. 3. 4. 5. 6.
7.
W W m' ¢ 0fI& 4 3 &-21 #- &-21 0fI& W ' (87! % m 0f /
1 ; 1 -; 0f9
& &-21 ] ' " W ( 7! &-2 1 0f;/ '#" - &-21 0f;/ 7 ( 7! ' 7 f G D f;/ H
F ; & 2 1 W W m m m'
9 G H DM &-21 F 3 -;&-21 V- &-21 F G W "' m m ] ' ED F GIH 3 " - &-21 F G 7 7 m m D &-21 F GIH , )
W '6(*7X! % m G H 4 3 ¢ -21 - 1 -; ) 8. EDM &-21 F 9 &-21 W W W W 43 AfI AfI& 9. D fI& H W W 3 ' & 8 3PfI 3 W W W W m m AfI -;&-21 F G 10. D &-21 H W W. 3 '%& 8 3 f -;&-21 + W W ( m ! ' 7 7 _
H G 0 f 11. D / F 3 f9
G ¤ F L m
= '8>?7A@ B = '8>?7A@ B = '8>?7A@ B = '8>?7A@ B = '8>?7A@ B = '8>?7A@ B = '8>?7A@ B m ,
= ' > 7A@ B
£ ¤ -;&-21 / fI
m
¤ -;&-21 ) ,
! #
13. 14.
15. 16.
17.
18.
19. 20.
21.
22.
23. 24.
" (
W 7 _ m ¤ m ' 7 , ED &-21 F IG H F G -;&-21 ) 3 W W 7 D &-21 V- D 0f;/ HOH F G f -21 0f;/ ¤ W W m ED a 1 - PED &-21 F GIHOH F 7 G f9 -;&-21 F m G ¤ W W 7 D a 1 k -21 f / H F G 3Pf9
0f / ¤ W m ED &-21 ED &-21 F GIHOH F 7 G 43f #- W F m G ¤ W W W W 3 ' & 8
H ED 0fI& 3PfI -;&-21 / £ fI ¤ W W. W W m ¤ m 3 '%& 8 ED &-21 F GIH f -;&-21 + -;&-21 ) , W W "' £ £ ¤ D &-21 0f;/ H -21 A - &-21 f / W m ED -21 F GIH "' ¤ 3 ¢ £ -;&-21 W + . - &-21 ) f * , W W D -21 V- D &-21 0f;/ HOH 43V -; £ -;&-2 1 0f / ¤ W W m ED a 1 - ED -21 F GIHOH 43V -; £ F m G ¤ W ED 1 &-2 1 f;/ 0 H ¡ = '8>?7A@ B W W D a 1 - D -21 0f / HOH f9 f / ¤ W
12.
m
P
W
25.
ED &-21 - ED &-21
W
:1
m F G9H H f -;&-21
m F G ¤
1.6. The Fresnel Integrals S(z) and C(z) 1.6.1. Derivatives with respect to the argument
fI& 0 " m ') (: 7! % 1 -; D 1 -; 43 AfI& <3 - &-21 m 2. EDM &-21 gF GIH 43 ¢ " m ') (: 7X! % -!&4 D R 1 -; F]3 m G 3 #&-21 1.
0 fI& ') (:7! % " m 1 -; D 1 -; 43 AfI& &-21 m 4. DM &-21 F G9H 7X! % 43 ¢ " m ')(: -!&4 D R 1 -; F]3 &-21
1 -; H &-21 AfI&
3.
# -
m 1 -; F GIH &-21 ' >?7R@CB =*
1 -; H &-21 AfI&
m G
R
= '*>h7A@ B
= '*>h7A@ B
m - R & 1 2- 1 -; F IG H > 7R@CB = '*?
1.7. The Generalized Fresnel Integrals S(z, ν) and C(z, ν) 1.7.1. Derivatives with respect to the argument 1.
2.
3.
fI ; ') 0 3 (: 7X! % f# -; -; 3AfI& 3 - -; AfI& &-21 &-21 m D &-21 F GIH 43 ¢ &-21 ') (: 7! % f V - -21 D R -; F3 m G 3 - R ;- F m &-21 &2- 1 0fI ') (:7X! % 3 f# -;
&;- -2 1 3AfI&
,9N
-
&;--2 1 AfI&
= '*>?7R@CB G9H = '*>?7R@CB = '*>?7R@CB
4.
m EDM & -21 F G9H 7! % m 43 ¢ &-21 ')(: fV - 2- 1 D R -; F 3 G 3 & -21
m - R & -;-2 1 F 9G H ' >?7R@CB =*
1.8. The Incomplete Gamma Functions γ(ν, z) and Γ(ν, z) 1.8.1. Derivatives with respect to the argument
fI& 0 3 ¢ fV -; - & -;-2 1 fI& m 3 ¢ 3 ¢ f# - -21 -R -; F m 2. EDM &-21 F GIH &-21 ¢ < ¤ 43 - -; fI& 3. - fI& 0 m -21 F < m G ¤ a 4. EDM -21 F G9H ¢ 3 3PfI )3 < fI& ¤ 5. fI& 0 m ¢ m 6. DM &-2 1 R F G9H 3 f -;&-21 R F T3 _ 7 m ¤ ' % f N '65_5_ !7 L 7. &- fI& 0 1 1J m 43 ¢ ' % f# -;&-21 N '65_5_7 ! 7 8. D -21 R F G9H ` 1 1^ a - P - fI& f fI& ¤ 9. m f -;&-21 a 10. ED &- -R PED -21 R F GIHOH
= '8>?7A@ B = '8>?7A@ B
1.
11. 12. 13. 14. 15. 16. 17.
G
G ¤ ¤
m F G ¤
fI& 0 ; 43fI fI& ¤ m m ED &- R PED a -21 F GIHOH 3fI -;&-21 R F G ¤ -21 - < &- fI& 0 43 ¢ ¢ 3 -;&-21 fI& ¤ m m ED &- a 1 -R PED -21 R F GIHOH 3 ¢ ¢ 3 ; F G ¤ &- V- fI& 0 f ¢ 3 - fI& ¤ m m ED a -R PED &-21 R F GIHOH f ¢ 3 -;&-21 F G ¤ fI& 3Q 3 ¢ f V -; - & -;-2 1 fI& = '8>?7A@ B a
< -
,
18. 19. 20. 21. 22. 23. 24. 25. 26.
L1 ,
m ED &-21 F G9H - fI& m ED a -21 F fI& 0
43 ¢ &-21 3 ¢ fV - -21 -R -; 0fI& = '8>?7A@ B &-21 ¢ _ ¤ 3 - -; - fI& m GIH -21 F < G ¤ ¢ 3 ; 43fI T3 _ fI& ¤ m m D &-21 R F GIH ¢ 3 ; f -;&-21 R F T3 < G ¤ '65_7 m &- fI& ¢ 3 ; f J 5_! 7 L ¤ '65_7 m D -21 R F GIH 3 ¢ ¢ 3 ; f -;&-21 ^ 5_! 7 ` ¤ m m ¤ ¢ _ ¢ 3 fI& 3 " 1 -; - A &-21 F G m m m . ED &-21 F ¢ 3 < GIH " -!&4 -R + &-21 F O G ¤
1.8.2. Derivatives with respect to the parameter 1.
] ]!
! ( 5 7 5_7 L ¤ & 63 N J _
1.9. The Parabolic Cylinder Function Dν (z) 1.9.1. Derivatives with respect to the argument 1.
2. 3. 4. 5. 6. 7.
m ' m fI& 0 F]3 G J $ L £ 43 ; &- F :G - 0fI& ¤ F3 )m G ' $ L 3 £ A F m G a fI& ¤ & J W W W W D fI& H 43fI 43 -; fI& ¤ W W W W D - fI& H 3PfI - a 0fI& ¤ W W. W W. m m D &-21 + F GIH f 3 ; -;&-21 + -; F G ¤ W W. W W. m m D &-21 - + F GIH f -;&-21 - + a F G ¤ W W D &- -21 A f / H £ -; 3 ; - -21 A - 0f / ¤ , ,
W . 4 3 a 8. EDM + -21 - A 0 f;/ H W . m + F G9H 9. D 3 £ -; 43 W a . - + . F m G9H 10. ED - + 1 £ -;
£ -; I+ -21 . - W A a 0f;/ ¤
W . m -; a + - F G ¤
W . m . -;&- + a 1 - + a F G ¤ W 3 £ -; -;&-21 V- W A ¤ H 11. D -21 V- A ; f / - W . m £ -; -21 - W + . ¤ 9 G H 12. ED &-21 - + F - W 3 ¢ £ -;&-21 O/ -;&-21 F f " G ¤ H 13. ED -21 - A ; f / - &-21 W . m £ -;&-21 O/ F m G ¤ 9 G H F 14. ED - + - &-21 W 15. ED A 0f / H W '65 $ ! 43 ¢ A & -21 '%&" f &- 3 ; ) ' ( ! ! $ 9 $ % &- -; a 0f;/ = '*>?7R@CB W 16. ED V- A f;/ H '65 $ ! 43 ¢ - W A & -21 '%&" f &- a f;/ = '*>?7R@CB ) ' ( ! ! $ & $ % & W W a 17. D 1 V- A D -21 A 0f;/ H H W m F 7 ( G , - A 0f / ¤ ) W . W . m 18. D &-21 - + P D &-21 + F GIHOH W . m m F 7 ( G , -;&-21 - + F G ¤ ) W W 19. D &-21 - A D A 0f / HOH F]3 G m -21 - W A f;/ ¤ , )
,93
20.
21.
22.
23.
24.
25.
26.
27.
28.
W . W . m a ED 1 - + PED &-21 + F GIHOH W . m F]3 G , -;&-21 - + ) W W D &-21 A D - A 0f / HOH W 7 m F 5_ G 3 , -21 A ) W . W . m D a 1 + D &-2 1 V- + F GIHOH W . 7 m F 5_ G 3 , -;&-21 + ) W W ED a 1 A ED -21 #- A 0f;/ H H F ¢ G 3 m W A , ) W . W . m ED &-21 + ED &-21 - + F GIHOH F ¢ G 3 m -;&-2 1 W + . , ) W W ED a 1 - A ED &- -21 A f;/ H H 3# -; 3 I+ -21 . -; - W A W . W . m D &- -21 - + D + F GIHOH 43V -; 3 ; - + a 1 . - W + .
W W . a D - -21 A D + -21 - A f / HOH 43# -; ¢ -;&- -21 W A W . W . m . a a a
+
+
ED 1 EDM - 1 V- + F GIHOH 43# -; ¢ W + .
1.9.2. Derivatives with respect to the order 1.
P1 ,
m F G ¤
f;/ ¤
m F G ¤
0f / ¤
m F G ¤
m F G ¤
£ -;&-21 - W F G
W #$ 7 #7 7 ! Y 3 C 3 £ F 3 G F G 3h N
&
,!5
M ]!
f;/ ¤
f / ¤ m F G ¤
2.
W 7 ¢ £ 3 4 3 & -21 - $ &-2- 1 ) , W " W #$ 1 $ W L Y 3-" 1 - 3 , J
N 1 $ !5
1 -21 ) 1 1 J L" W " W ' ( L 7 7 ¢ £ 3 3 &-21 F G - J $ L J F 3 1 J L" W M ]! £ -;&-!&4 - a
a 1 F G W #$ 1 7#7 7 ! Y C £ 3 F3 3 G 3 F G N
&
7 ( " W ¢ £ £ ¢ $ (:7! % 'T( $ 5_7X! % F 3 G 3 -;&- -21 - W 7 1 ¢ £ £ ¢ -21 -!&4 3 &-!&4 - $ W &- ) , &- a 1 ) , 1 $ #$ W "" ! Y "" N
3 / 1 -2 1 - ) 3 , 1 L)" 1 1 $ 5 1 J
W ¢ £ 3 43 &-21 / -;&-21 ) 3 , '65_7 W #$ W W '& ' % ¢ ! a 1#- 1 N 1 'g5 3 V- F]3 3 3 & & J L'
7 #7 #$
&
& & N & ! 1 -1
& £ £ £ &4 C -21 ) , 1 ) , £ £ £ £ 3 C 1 ) , 83 C 3 -2 1 ) , =
G ¤
7 G ¤
3.
1.10. The Bessel Function Jν (z) 1.10.1. Derivatives with respect to the argument 1.
m ' fI& 0 F G 3 ¢ J $ L
, ;
¤
2 0fI&
@CB
+ (
2.
3.
4. 5. 6. 7. 8.
9.
" % $ m ( ! ! % a 0 fI& ¤ m ' -;&-21 43 ¢ J $L 2 F G ¤ m ¢ ' $ L F G ¤ F G f;/ J &. I+ -; 2 f / ¤ 7 m ¤ F G &-21 f / ( 7! ' m ¤ F G &-21 0f /
" 43& -; ')( (m ] $ ! ! % &
EDM &-21
m m F GIH F G
7 f;/ 0 F]3 G m f / 0 F G
. I+ a 1 a 1 0f / 0 . I+ a 1 -;&-21 0f / . - + a & a 1 0f;/ / *F m G a 1 a 1
f / 0 F m G + -!&A . & -21 fI - $&% '6 ')5 ( $ $ ! !
10. 11. 12. 13.
15.
m m F G -;&-21 F G ¤ -
2
0f /
= '*>?7R@CB F m G ¤
7 ' m ¢ ED &-21 F IG H J $ LF G F G &m m m . ED -21 F GIH F G - + -21 F G ¤ ( 7X! ' m m m ¤ . ED I+ &- a 1 F GIH F G &-21 -;&-21 F G 7 m m m . ED I+ &- -;&-21 F GIH F G &-21 -;&-21 F G ¤ m . ED I+ &-21 a 1 F GIH m ¤ 43 ¢ / 8F m G a 1 -21 a F m G G F 1
-;&-21
m ! '#" &-21 &-21 fI& 0 &-2 1 = '8>?7A@ B m
14.
N1
,9:
16. 17. 18. 19.
20. 21. 22. 23.
24. 25. 26.
27.
( 7! ' £ = '8>?7A@ B D -21 R &-21 F 9G H If &-21
- R m ! '#" &-21 fI& &-21 fI& 0 &-21 £ fI& ¤ m ! ']" &-21 0fI& &-21 fI& 0 &-21 £ fI& = '8>?7A@ B m ! '#" ¢ a &-21 fI& 1 -; fI& 0 3 1 &-21 £ fI& = '*>?7A@ B m ! '#" ¢ &-21 0fI& 1 -; fI& 0 3 &-21 £ fI& ¤ m ¤ m ( 7X! ' £ m D -21 &-21 F GIH fI &-21 - m m m ( 7X! ' £ ED -21 &-21 F GIH = '8>?7A@ B fI &-21 - ' m 0f;/ F G 43 ¢ J $ L a f;/ -; a 0f;/ ¤ m ']" . £ &-21 & -21 0f;/ = '8>?7A@ B I+ &!- & &-21 f;/ m ']" . &-21 1 -; 0f / 3 I+ &-!& &-21 £ f / = '8>?7A@ B &-21 &-21 0f;/ 1 -; 0f;/ '#" . m £ '*>?7R@CB I+ &!- & 1 -; f;/ = m '#" . £ ¤ &-21 0f / a 1 0f / I+ &-21 a 1 f / ( 7X! ' m m . a D -21 & -21 F GIH
+ f F & 2 1 1 &-21 G '*>?7R@CB = % ' & ( X 7 ! m m . D -21 1 -; F GIH f &-21 - + a 1 &-21 F G = '*>?7R@CB m m D -21 &-21 F G 1 -; F G9H ' m ¤ ( 7X! f &-21
- + a 1 . 1 -; F G
m
28.
29.
30.
, <
+ (
31.
32.
m m ED -21 &-21 F G a 1 F G9H ' ( 7! m ED &-21 F GIH F3 m G -; -21 43 ¢
N1 ,
m . f &-21
- + a & a 1 F G ¤
'
J $L
m a F G
F m G ¤ a ;-
1.10.2. Derivatives with respect to the order 1.
2.
3. 4.
"" ' ' % ¢ 0& ¢ & -21 $&J % ' L ( $ ! 0 & ¤ £ & <3 £ & 0 = = @ B V C7 !C@MB * 1
£ & £ & 0 = = @ B V C7 !C@MB * -21
£ & a 0& 3 43 ¢ £ & a " 1
-;&-21 & " 1
' % ' % & -21 L
L J ' ( $ ! a 1 0& 3 F G ')J ( $ ! % $ & $ % 1 £ ¢ £ ¤ % &- a 1 & -21 & <3 3 &- - -;&-21 0& 1 - & 0
M ]! M ]! M ]! M ]!
Y 2- 1 M ]!
£ & 1 -; & <3 3 ¢ £ & &-21 & ""' 1 -; ' % & -21 ( L ' % " 3 $&J % ')
( $ ! 1 - & <3 ')( $ ! % $ 1 ")"'& £ Y -21 % 3 ¢ a 0& -; 1
-21 & 3 3 ¢ a &- -21 & 1 M ]! ¢ &-21 £ a 1 / -;&-21
6. " 1
')( $ ! % L 7 G 3 F 3 Y &-21 J
X D F F ) ' * ( ! 5 * ( 6 ' 5 $ % $ 1L $ $9% 1L J J ,9L
5.
£ & 0 ¤ 7 GG
Y
¢ £ &-21 / -; a
Y DXF F
£ & H " ) ' ( : ( X 7 ! $ % L & -21 1 &$ % ')(* $ (:7! % $ 5 J $ * ( '65 &L J J 7
G 3 F 3 GG £ 3 & £ & H
£ &
1L
= '*>h7A@CB
1.11. The Bessel Function Yν (z) 1.11.1. Derivatives with respect to the argument 1.
2.
m ' 0fI& F G 3 ¢ J $ " 43& -; ')( (m ]$ ! ! % &
3.
m m D &-21 F GIH F G
4.
5. 6. 7.
m f / 0 F G . I+ a 1 a 1 0f;/ . I+ a 1 -;&-21 0f / . - + a & -;&-21 0f / 43 ¢ a 1 / 8F
( 7X! ' m F G &-21
0fI& ¤ m F G ¤
f / ¤
m a m m O G 1 a 1 F G ;- &-21 F G ¤
f / 0 F m G I+ -!&A . & -21 If - $9% '6 ')5 ( $ $ ! ! - 2 f /
m F m G I + -; . -21 F m G ¤ 9. D -21 F G9H ( 7X! '%& m m . &-21 -;&-21 F m I G H G F 10. ED I+ &- a F 1
8.
¤
2 0fI& " % $ m ( ! ! % a ' -;&-21 43 ¢ J $ L 2 . I+ -; 2 0f / ¤ 7 m 3 F G &-21 f;/ ¤
L
,9P
= '*>h7A@CB
G ¤
+ (
11.
. ED I+ &- . D + &-21
12.
13. 14. 15.
16.
17.
m F -;&-21 m F
-;&-21 3 / *F
7 m G9H F G &-21 -;&-21
m F G ¤
G9H
m a 1
m G -21 a 1 F G ;- &-21
m ! '#" ¢ a &-21 1 -; fI& 0 3 1 &-21 m m ! '#" D -21 R 1 -; F GIH 3 - R m ']" . ¢ a &-21 1 -; 0f;/ 3 1 I+ &!- & 1 -; £ f;/
m F G ¤ = '8>?7A@ B = '8>?7A@ B = '*>?7R@CB
&-21 &-21 0f;/ 1 -; 0f;']/ " . 3 ¢ a 1 m '8>?7A@ B £ I+ &-!& & -21 f;/ = &-21 0f;/ a 1 0f;/ '#" . 43 ¢ a 1 m £ ¤ I+ &-21 -;&-21 f;/ m m m '#" . ED -21 1 -; F GIH 3 - + a 1 1 -; F G = '8>?7A@ B m m D -21 &-21 F G 1 -; F G9H ']" m ¤ . 3 m - + a 1 &-21 F G m m ED -21 &-21 F G a 1 F G9H m ¤ 3 7 f &-21 - + a & . F -;&-21 G &-21 &-21 0f / 1 -; 0f ']/ " . 3 ¢ a 1 m £ '8>?7A@ B I+ &!- & &-21 f / =
18. 19.
20.
21.
22.
&-21 1 -; 0f;/ & -21 0f;']/ " . 3 ¢ a 1 m £ I+ &-!& &-21 ;f / 3 N
= '8>?7A@ B
, 3 8 , 3 8
23.
24.
25.
26.
&-21 &-21 0f;/ &-21 0 f;/ ( 7X! ' . f &-21
I+ &-!& 1 -; &-21 0f / -;&-21 0f / ( 7X! ' . f &-21
I+ &-21 a 1
£ f / ¤ £ f;/ ¤
a 1 0f;/ 1 ;- 0f;/ '%& ( 7X! f &-21
I+ &-21 . a £ f;/ ¤ 1
m m D -21 &-21 F G &-21 F G9H ]' " m ¤ m a . - + 1 1 -; F G m m ED -21 1 -; F G &-21 F G9H ']" m ¤ 3 m a . - + 1 &-21 F G m ¤ m m m '#" a . D -21 &-21 F G -;&-21 F GIH - + & a 1 F G m m ED -21 a 1 F G 1 -; F G9H ']" m ¤ . 3 m a f
+ F & 2 1 & a 1 G
27.
28. 29.
1.11.2. Derivatives with respect to the order 1. 2.
" "' ' M ]! % 3 ¢ & ¢ & -21 $&J % ')L ( $ ! & ¤ M ]! 3 0& 43 ¢ M ! ¤
1 1
2<1
1.12. The Hankel Functions Hν(1) (z) and Hν(2) (z) 1.12.1. Derivatives with respect to the argument 1. 2.
. m . + f / 0 F G I + -;
. m m EDM -21 + F G9H F G I + -;
31
. 0f / 2 . m . -21 + F G +
= 7] X@CB = 7] X@CB
+ (
3. 4.
. I+ a 1 DM &-21
. ( 7X! m . U + a 0f / F G & -21 + -21 1
. . + 1 0f;/ + f;/ H ¡ 1 -; 1 -;
!, ,
= 7 ] X@CB = '8>?7A@CB
1.12.2. Derivatives with respect to the order 1.
2.
3.
4.
" "' . ' % & -21 L . + & $&J % ')( $ ! + 0 & = 7] X@CB "" ' ' % ¢ a + . & <3 3 ¢ & -21 $9J % ')L ( $ ! + . & = 7 @CB 3 8 M ]! * Y 1 + -2 1 . 3 ¢ a 1 £ & <3 £ & 43 ¢ = 7 @CB 3 8 M ]! * Y -2+ -21 1 . £ & 3 ¢ a 1 £ & 0 3 ¢ = 7 @CB
3 8 M ]! 3 ¢ 3 8 M ]! 3 -;
1.13. The Modified Bessel Function Iν (z) 1.13.1. Derivatives with respect to the argument 1.
2.
m fI& 0 F G J 43& -; ')( (m
3.
m DM &-21 F GIH
4.
f;/ 0 F]3
5.
6.
f;/ 0 . I+ a 1 a 1
'
¤
2 0fI& ]! " m ]! " $ ! % &- % $ ( ! % - a 0 fI& ¤ m m ' F3 G -;&-21 J $ L 2 F G ¤ 7 ' m ¤ G G F]3 G ;f / J $L F & F m G I+ -; . 0f;/ ¤ 2 7 m f / 0 F G &-21 0f / ¤
$L
3,
7 m . I+ a 1 -;&-21 f / 0 F G &-21 0f / ¤ a . 8. - + & a 1 ;f / 0 m ¤ / 8F Om G a 1 a F m G F 1
-;&-21 G 9. k 0f / F m G I+ -!&A . & -21 3PfI - $&% '6 ' 5 ( $ $ ! ! - f / = '*>?7R@CB 2 m m m 7 ' $ L F G F G ¤ G 10. D &-21 F G9H F 3 J &m F]3 m G - + . -21 F m G ¤ 11. ED -21 F G9H ( 7! ' m m . &-21 -;&-21 F m G ¤ I G H G F F 12. ED I+ &- a 1
( 7X! ' m m . &-21 -;&-21 F m G ¤ I G H G 13. ED I+ &- F F -;&-21
m . 14. D I+ &-21 a 1 F G9H m ¤ 3 ¢ / *F m G a 1 -21 a F m G F G 1
-;&-21
m ! '#" 15. &-21 &-2 1 = '8>?7A@CB &-21 0fI& m ! " '#" -;&-21 £ ¢!!£ fI& ¤ 16. k &-21 - a 1 fI& 5n'65 5n'65 1 L £ J 17. k &-21 fI& fI # -21 5_ 7X! 1 N 1 5_7 m 1 ¤ ! ' ( X 7 ! m £ fI &-21 - R '8>?7A@CB I G H F = 18. ED -21 R
&-21 m ! " '#" £ ¢) m ¤ m ¢ G F 19. D -21 V - R a 1 H 43 7 m 20. ED R F GIH 5n'65 #$ 5n'65 1 ¤ 3 ¢ £ fI # - -;&-21 J 5_ 1 7XL! N 1 1 5_7 ! 7.
3 3
+ (
21. 22. 23. 24. 25.
26. 27. 28.
29. 30.
m ! ']" &-21 0fI& &-21 0fI& &-21 £ fI& ¤ m ! '#" &-21 0fI& &-21 0fI& &-21 £ fI& m ¤ m m ( 7! ' £ ED -21 &-21 F GIH fI &-21
- m m m ( 7! ' £ D -21 &-21 F G9H fI &-21 - m ' V f / F G J $ L a 0f / -; a 0f / ¤ m '#" . £ &-21 V& -21 f;/ 0 I+ &!- & &-21 f;/ m '#" . £ &-21 V1 -; f / 0 I+ &-!& &-21 f / &-21 &-21 f;/ 1 -; f;/ ']" 0 . m £ I+ &!- & 1 -; f;/ m ']" . £ &-21 0f / a 1 f / 0 I+ &-21 a 1 f / m ED &-21 F GIH F3 m G -; -21 ' $L a F m G a -; KJ ' ( 7! m m . D -21 V& -21 F G9H f &-21
- + a 1 &-21 F G ( 7! ' m m . a
+ ED -21 1 -; F G9H f F & 2 1 1 &-21 G
31
= '8>?7A@CB = '8>?7A@CB
= '8>?7A@CB = '8>?7A@CB = '*>?7R@CB ¤ F m G ¤
31.
= '*>?7R@CB
32.
33.
m m ED -21 &-21 F G 1 -; F' GIH m ( 7X! f &-21
- + a 1 . F 1 -; G m m ED -21 &-21 F G a 1 F GIH ' ( 7! f &-21
- + a & . a 1
34.
395
= '*>?7R@CB = '8>?7A@CB m
F G ¤
35. 36.
37. 38.
39. 40.
41. 42.
0fI& F m ED &- - R PED a &-21 R F GIHOH F 7 G 7 &- P - 0fI& F 3 a - P -
7 G £ fI - 0fI& ¤ m £ fI - ;- &-21 -R F G ¤ G 3 £ fI - fI& ¤
m ED a R ED &- 2- 1 V-R F GIHOH F 7 3 G 3 £ fI -;&-21 R F m G ¤ 7 £ &- - < 0fI& F 3 G If - - 0fI& ¤ m a I G O H H D #- R D &- -21 R F F 7 3 G £ fI -;&-21 -R F m G ¤ 7 a P - - 0fI& F G 3 £ fI fI& ¤ m a D &- R P D -21 -R F GIHOH 7 m F G 43 £ fI - -;&-21 R F G ¤
1.13.2. Derivatives with respect to the order 1.
2. 3. 4.
"' ( ' M ]! % L 3 ¢ a 1 & & -21 J % 'T
( &! 0& ¤ 7 M ]! £ £ @ B C7 !C@MB = = * 3 & #- & 1
£ £ ¤ * D & 3 & H M ]! 7 3 £ & a -;&-21 & a 1 " & 0
1 ' ( 7X! ' % & -21 ( 7X! 3 £ & a 1 0& 643 ¢ $9% ')( $ ! F ,G -; a 1 & " ( L ( ! ' % -21 3 " ')J ( $ ! % $ % £ Y 3 ¢ a 1 & &- 1 ¢ -; &-21 0& -21 £ & 3 43 &- &- a 1 & -21 & 1 - £ & 0 ¤ 3;
+ (
5)
( 7! ' £ 5. & 00 ;- &-21 & a 1 " & 0&3 a 1 & ' ( 7X! Y £ & £ & 43 ¢ % & -21 $&% ')( $ ! F G -; a 0& 1
" ( L ( ]! ' % -21 3 " ')J ( $ ! % $ % £ Y 3 ¢ a &1 &- 1 ¢ -; &-21 0& -21 £ & 3 43 &- &- a 1 & -21 & 1 - £ & 0 ¤ M ]! 7 £ & <3 £ & 00 6. -;&-21 & a 1 & 0 1 -; ( 7! ' 3 " "' £ & £ & &-21 & " ' % & -21 ( L ' %
-21 L 3 $9J % ')
( $ ! 1 - & " ')J ( $ ! % $ % 1 £ Y 3 ¢ -21 &- -21 ¢ & -; a 1 0& -21 £ & £ ¤ 43 &- &- -21 & -21 & 1 - & 0 M ]! 7. " 1
" ')( $ ! % ( 7 ! £ &-21 O/ -;&-21 & -21 J $ L (*'65 ) ' * ( ! 5 9 $ % $ % $ 1 1L J L J
7 7 Y D £ F F G 3 F 3 G;G 3 £ & V- £ & H " " ')( $ (:7! % ( 7 ! & -21 £ J $ L (*'65 &!- &4 / -; a 1
) ' * ( : ( X 7 ! 5 9 $ % $ % $ & 1L J L J
7 Y D £ F F G 3 F 3 GG 3 43 £ & - £ & H ¤ 7
£ & 3
1.14. The Macdonald Function Kν (z) 1.14.1. Derivatives with respect to the argument 1.
2.
m ' fI& 0 F]3 G J $ L 2 0fI& ¤ " " 43& -; ') m ( ]! $ ! % % ( $ m ( ! ! % a 0 fI& ¤ &- 3 :
$
m m m ' F GIH F G -;&-21 J $ L 2 F G ¤ F3 m G I+ -; . ¤ 4. f;/ 2 0f;/ 43 ¢ '% & f &-21 - ¤ a . ; f / 0 5. I+ 1 a 1
( 7! ' m a . a 1 -;&-21 a F m G ¤ G 6. - + & a ; f / 0 F 1
1
7. f / F3 m G I+ -!&A . & -21 f - $&% '6 ')5 ( $ $ ! ! - '*>?7R@CB 2 f / = m F m G - + . -21 F m G ¤ 8. EDM -21 F GIH m ¤ m . 7 F m G a 1 -21
9 G H 9. DM + &-21 a F F G a 1
1
¢ £ £ ¤ 10. k &-21 a 1 0fI& 3 / 6 fI -;&-21 -;&-21 SEa 1 )SEa fI& 11. k 1 3 ¢ SEa _ / g £ fI &- S -21 &- S -21 £ fI& = > 'I@ B S -; 12. k &-21 )SEa fI& 43 ¢ S 1 / 6 £ fI - S -21 - S -21 - S -21 £ fI& = > 'I@ B S -; S fI& 13. k - -21 )SEa 3 ¢ SEa 1 / g £ fI - S -21
- S -;&-21 - S -;&-21 £ fI& ¤ S SEa 1 - )SEa fI& 14. k 1 3 ¢ SEa _ / g £ fI &- S -21 - &- S -21 £ fI& ¤ S 15. k &-21 - )SEa 1 fI& 43 ¢ S / 6 £ fI - S -21 - S -21 - - S -21 £ fI& ¤ SEa S 16. k - -21 # - )SEa fI& 3 ¢ S 1 / g £ fI - S -21 - S -;&-2 1 #- - S -;&-21 £ fI& ¤ SEa m 17. D -21 R )SEa 1 F GIH 43 ¢ SEa / 6 £ fI - S -21 S -; - S -21 F m G = >_'I@CB S -; 3.
EDM &-21
3<
+ (
18.
ED
20.
D
21.
ED
22.
ED
24. 25.
26.
m F 1 IG H ¢ S / g £ fI - S -21 S - S -;&-21 S m &- S -!&4 R )SEa 1 F G9H 3 ¢ S ? / g £ fI &- S -21 -;&-21 &- S -21 F m G = S -; m -21 V-R )SEa 1 F GIH 3 ¢ SEa / g £ fI - S -21
S -; - R - S -21 SEa SEa &-21 - R )SEa F m G9H 1
3 ¢ SEa / g £ fI - S -21
S - R - S -;&-21 SEa m &- S -!&4 -R )SEa 1 F G9H 43 ¢ S _ / g £ fI &- S -21 -;&-21 V- R &- S -21 S
ED ES a & -21 R ) SEa 3
19.
23.
5)
28.
29.
> 'I@ B m F G ¤ m ¤ G F : m F G ¤
m ' 0f;/ 0 F]3 G J $ L a f;/ -; a 0f;/ ¤ . &-21 & -21 f;/ 0 43 ¢ / f &-21 + &-!& &-21 £ f;/ ¤ &-21 0f;/ a 1 f;/ 0 43 ¢ / f &-21
I+ &-21 . a £ f;/ ¤ 1
m ED &-21 F G9H F m G -; -21 ' $ L a F m G F m G ¤ a -; J m m . D -21 & -21 F G9H / f &-21 - + a 1 &-21 F G ¤ m m D -21 &-21 F G a 1 F GIH / f &-21 O - + a & . a F m G ¤ 1 &-21 &-21 f / &-21 f ']/ " 0 . m £ '*>?7R@CB I+ &!- & &-21 f;/ =
27.
m ¤ G F 8
3 L
30.
$
m m ED -21 &-21 F G & -21 F GIH ' m ( 7X! f & -21
- + a 1 . '8>?7A@CB F & -21 G = - &-21 fI& 0 7 3 ¢ F 3 G F 7 G -;&-21 - 0fI& ¤ m ED - R PED -21 R F GIHOH 3 ¢ F 7 3 G F 7 G -21 V- R F m G ¤ < &-21 - fI& 0 7 43 ¢ F 3 G F 7 G -;&-21 0fI& ¤ m ED R PED -21 - R F GIHOH 3 ¢ F 7 3 G F 7 G -21 R F m G ¤ &- S -21 - SEa 1 )SEa 1 fI& 0 3 £ fI - S -21 #- )SEa fI& ¤ 1
m ED SEa a 1 - R PED &- S -!&4 R )SEa 1 F G9HOH 43 £ fI S -;&-21 -R )SEa F m G ¤ 1
a E S a S 1 < - -21 - )5n SEa'!1 % fI& 0 43 £ fI SEa 1 )SEa 1 fI& ¤ % m ED &- S -21 R PED a2S -21 -R )SEa 1 F G9HOH 5n% '! % 3 £ fI - S -;&-!&4 R )SEa F m G ¤ 1
a E S a S 1 - P - -21 )5hSE'a ! 1 % fI& 0 % £ fI SEa 1 - )SEa fI& ¤ 1
m ED &- S -21 - R PED a2S -21 R )SEa 1 F G9HOH 5n% '! % £ fI - S -;&-!&4 V-R )SEa F m G ¤ 1
- &-21 )SEa 1 fI& 0 5n'! % (*'! % -;&-21 - )SEa fI& ¤ 1
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
3 P
+ (
42.
5) ,
m ED - R PED -21 R )SEa 1 F GIHOH '! % m 5n * ( '! % -21 - R )SEa 1 F G ¤
1.14.2. Derivatives with respect to the order
M ! ¡\¤ M ! 2. M ! 3. M ! 1 " 4.
1 M ! 5. &-21 " " ' ' % & -21 L J ' ( ! $&% $ 1.
"' ' % & -21 L ¤ J ')( &! 0& %
= = @ B C7O!C@MB " O 43 £ & £ & <3 £ & 0 ¤ 43 ¢ 43 £ & &-21 & 1 -; 0& " ( L ' % -21 0& 3 43 ¢ / ')J ( $ ! % $ &- -21 0& 1 !& -21 % -; a 1 0& -21 £ & ¤ 3 ¢ £ & <3 £ & 00 6. &-21 & 1 -; 0&" ""' ( L ' % ' % & -21 L ¢ J ')( ! ')J ( $ ! % $ &- -21 & $9% $ -21 & 3 3 / 1 !& -21 £ & ¤ % -; a 1 0&
2 1 M ! 7. " a 1
( 7! " ')( $ ! % L 3 ¢ a 1 £ &-21 &4
-;&-21 &-21 J $9% ')(* $ ! % $ 5 1 L $ (*'65 1 L J J 7 7
Y D - F F G 3 F 3 G;G 43 £ & H " ( 7! " ')( $ (:7! % L & -21 ¢ £ a a J 3 1 &-!&4 &4 -; 1
$9% ')(* $ (:7X! % $ 5 & L $ (*'65 1 L J 7 J
Y D - F F G 3 F 3 GG 3 3 £ & H ¤
5N
L
H
1.15. The Struve Functions Hν (z) and Lν (z) 1.15.1. Derivatives with respect to the argument 1.
2.
H fI& 0 " ' ( $ ( 1L m
& - ) F3 G $&% J ')(* $ ! % L
F]3 G $9% J ')(* $ ! % & - ' ( $ L F]3 G J & 5 1L 7 m a -21 m ¤
Y F nG H a 0fI& 3 F nG -21 J 5 Q( 5 1 L F m G J
m . H f;/ 0 F G I+ -; H -; f;/ ¤ m . - H f / 0 F3 G - + a H a 0f / $ 5 1L ( 7X! ' m a & 2 1 3 F G &-21 -21 5nJ ')( $ 5 F m G ¤ 1L J
m m . + a 1 H a 1 f;/ 0 F G &-21 ¤ m m m . EDM &- -21 H F G9H F]3 G - + a -21 H -; F G ¤ m m m . D I+ &- H a 1 F G9H 43 ¢ F G &-21 -;&-21 ¤ m m m . DM a -21 H F G9H F G + -; -21 H a F G 5 1L $ 7 m a ¤ &-21
& 2 1 J G 3 F -;&-21 5:' ( $ 5 F G 1L m J
L 0fI& " ( 1L m
& - ')( $ F]3 G $&% J ')(* G nG L - 0fI& ¤ L F ] F 3 $ ! % J &-
3. 4.
5. 6.
7. 8.
9.
5)
+ (
!; ,
" ( J 1 L & ¢ ')( $ L F 3 G 4 3 10. F 3 G $&% ')(* $ ! % J & - 5 7 1 Y F m G L a fI& F m G a -21 -21 J Q( L 5 F]3 m G 5 1L J
m . L 0f / F G I+ -; L -; f / ¤ 11. F m G - + a . L a f;/ 12. - L 0f;/ $ 5 1L 7 m a & 2 1 G J F G &-21 -1
5n')( $ 5 1 L F 3 m J
m m . ¤ a &-21 13. I+ 1 L a 1 0f;/ F G m F3 m G - + a . -21 L F m G ¤ 14. ED &- -21 L F GIH -; m . 3 ¢ F m G &-21 -I&-21 m I G H F 15. ED I+ &- L a 1
m F3 m G I+ -; . -21 L a F m G a 16. ED -21 L F GIH 5 1L $ ( 7X! ' m a &-21 F3 m G F G &-21 -;&-21 5nJ ')( $ 5 1L J
¤
¤
¤
¤
1.15.2. Derivatives with respect to the order 1.
2.
H M ! 7
&
H M ! 3 7 & -21
J
1 J L
1
3 0&
!¡!¡
]' " "' &
& L)" &- -21 F G D '])" " ' % & -21 ¢ 3 5,
> @CB = Q
<!¡
7 3 F 3 GIH "" ' L > @CB $&J % ')( $ ! H- & = Q
3.
4.
5.
M ]!
3 ¢ a 1 0& #' " ; - "' ¢ 4 3 &
! _!¡
H
L
H
"" ' ' % & -21 ( L 3 J$&% ' ( $ !
H M ! 1 * Y C £ & 3 £ & 0
£ & 3 £
H
- & = Q> @CB
0& = = @ B # O !C@CB
H M ! *
1 Y £ & 3 £ & 0&3 £ & <3 £ & 0 = = @ B # !C@CB H M ]! £ & 3 £ & 0 a 0& a 1
£ 1 £ ¢ 3 & 3 0& -I&-21 0& H a 1 & <3 a 1 & 0 7 a 7 7 £ £ F G 1 F G D C F 3 GIH " ' % & -21 ( L ' % a & -21 1 L)" 3 F G $&J % ' ( $ ! - -21 0& 3 F G 1 $9% J ') ( $ ! ']" L & -21 J 1 L" ¢ J ') ( $ ! % F G 3 3 " -21 J L &- -21 J L &
1 ; ' ( ! 3 / 8F G $&% $ % £ £ Y 43 ¢ a 1 a 0& 1 ¢ 1 - & <3 -21 1 £ - & 3?43 - -21 & -21 0& 3 -21 -21 £ & ¤ ¢!¢ 7 L M ! = Q> @CB \¢)\¢ & <3
6.
7.
8.
9.
3 & <3
L M ! & <3 7
¢)\¢! \¢)\¢!
53
= Q> @CB
¢)\¢ \¢)\¢
¤
+ (
10.
11.
12.
13.
14.
15.
16.
3 0& 3
!; ,
¢)\¢) \¢)\¢)
¤
( O! ']" "' 0&
¡
7 1 J L " F G &- -21 D 3 F 3 G9H 1 J L '#)" " ""' ' % & -21 ( L J % ') ( $ ! L 0& = Q> @CB 9 $ ( ! '#" "' L M ]! ¢ 3 &
¡\ !
; - ""' ' % & -21 ( L 3 $&J % ')
( $ ! L- 0& = Q> @CB 7 7 L M ]! £ £ 0& 3 F
C ,G - & <3 * 1
3 43 £ & <3 £ 3P& = = @ B C7 !C@MB
L M ]! 4 3 ¢ 7 & -21 3 43 ¢
L M ]!
L M ]!
1
-1
*
*
£ & 3 £ 3
0&
£ & <3 £
£ & <3 £
& 0 3 £ & <3 £
3
L M ]! £ & 3 £ & 0 a 0& a 1
1 £
£ 3 & <'%3 & & 0 -;&-21 0& L a 1 & <3 ( 7X! a 7 F G 1 F G D C £ £ F " ' % & -21 ( L J ')( ! F3 G $9% $ - -21 & ¢ ' % a & -21 $9% J ') 1 L ( " $ ! 43 F G 1
5 5
& 0&3
0& 7 ;- &-2 1
3 IG H
C
¤
& 0 ¤
J
'#"
E
& -21 ¢ J 1 L " ¢ ') ( $ ! % F G 3 3 " ( L ( &-21 J &- -21 J L ¢
1 ; , G ' ( ! % 43 / 8F $&% $ Y a & & <3 £ -21 £ & 1
1 1 3 - -21 0& -21 & <3 £ -21 -21
L M ]! £ & <3 £ & a 0& 1
-I&-21
£ £ & <3 & 0 -;&-21 & "" ' " ( ' % & -21 ( L ' % L -21 ( ! 3 $9J % ')
( $ ! a 1 & / ')J ( $ ! % $ % 1 £ £ Y a & 1 - & <3 -21 &
&- 1
2 1 3 -;&-21 0& £ 1 - 1 - & <3 1 -
L
J
17.
£ & ¤
£ & ¤
1.16. The Anger Jν (z) and Weber Eν (z) Functions 1.16.1. Derivatives with respect to the argument
" ( 1 J L & - ')( $ ; L F J 0& F 3 , G $9% ')(* $ ! % J ! Q( Y J & <3 3 ¢ -21 F " ( 1L & - ')( $ ]F 3 ,G -; $&% J ')(* G < L F 3 $ ! % J
1.
2.
Y 3. 4.
F 7 G
J 0f;/ 3
G \ \ ]F 3 ,G ( 5 7 G F G
\ \ F , G 5 5_7 G F]3 G
¤
! -21 ( ¢ ¤ J a 0& 3 4 3 F ¢ ' ¤ , $ L 4 3 J &
2 J m . F G + -; J -; 0f;/ ¤ m ! . & -21 ' ( $ ( 5 7 G G F3 m I+ -;&-21 F F m G
5;
+ (
:1
m - J 0f;/ F 3 G - + a m ! 3 F G m - + a a
. J a f;/ ¤ . & -21 ')( $ 5 5_7 G F]3 m G 1 F m F3 m G - + a . -21 J F m G 6. EDM &- -21 J F GIH -; ! ¤ m . & -21 ')( $ ( 5_7 a a G F m G 3 F G m - + 1 F m F m G I+ -; . -21 J a F m G a 7. D -21 J F GIH ! ¤ m . & -21 ')( $ 5 5_7 G F]3 m G 3 F]3 G m I+ -;&-21 F " ( J 1 L & - ')( $ ; ) ' ( G G 9 $ % 3 L 3 8. E & 0 F F F $ ! % \ \ G J 7 Q( ( 5_7 ¤ Y E & -21 3 ¢ 3 ¢ XF G G F " ( J 1 L & - ')( $ ; G L F]3 ,G $&% ')(* F 3 F 9. $ ! % \ \ ,G J 7 ( 5 5_7 ¤ Y E a & -21 ¢ 3 ¢ a CXF G G F F 7 G 43 ¢ ' $ L E & ¤ 10. 2 J F m G + -; . E 0f;/ 7 F m G + -;&-21 .
11. E f;/ 0 m -; ) ' ( ( $ 5_7 Y & -21 3 ¢ 43 ¢ XF G F m G ¤ 5.
12.
- E f;/ 0 F]3 m G - + a Y & -21 3 .
7 m . a 0 f;/ m F G - + a a 1
¢ 3 ¢ XF ')( $ 5 5_7 G F G ¤ m
E
5:
13.
14.
J ν (z )
m m F G9H F]3 G - + a m " 3 '&(' & F3 G m Y & -21 3 ¢ 3 ¢ m ED a -21 E F G9H F m G I+ -; . -21 E a F m G Y & -21 3 ¢ 43 ¢ D &- -21
Eν (z )
. -21 8
E
XF
E
m F ;- G
')( $ ( 5_7 G F m G ¤
m 3 " ']" 8 F3 G m ')( $ 5 5_7 ¤ G F m G XF
1.16.2. Derivatives with respect to the order 1.
2.
3.
4.
J M ! 7 & -21 3
' % & -21 1 J L" 1 J L '#)" "
#' ")"
L $9J % ')( $ ! 0&
&
H
( 7X! ' & -21
& 2 1 F , G 3 '])" " ' J M ! % 43 ¢ &-21 & -21 $&J % ')L ( $ ! & -I ( 7X! '
E M ! 0& ']")" ' % & -21 L J ')( ! $&% $ 3 H 0& 7 & -21 ¢ ')( $ ¢ 3 3 XF ']")" ( L ' E M ! % &-21 $&J % ' ( $ ! H -I 7 & -21 ¢ ')( ¢ 4 3 43 XF 5<
7 2- 1 5_7 G
¢ F ')( $ 5_7 G F G ¤
-; & 5 7 ¤ & -21 ')( $ _ G F G F H
1L J
F ,G - -21 1 )L "" J
a -21 7 5 ')( $ 5 7 ¤ F]3 G 1 n
& 3 ¢ & a -21 7 $ 5_7 G F G 1 5n')( $ 5 7 ¤
+ (
!<
1.17. The Kelvin Functions berν (z), beiν (z), kerν (z) and keiν (z) 1.17.1. Derivatives with respect to the argument 1.
2.
3.
4.
5.
6.
7.
f;/ 0 F m G I + -; . D
F
'
f;/ 0 ' m . G + -; D
2
0f;/ <3
2
0f;/
. I+ a 1
. I+ a 1
. I+ a 1
. I+ a 1
' '
7 m a 1 f / 0 3 F G &-21
O')(:7X! Y D F f " G FOf " G O ')(:7! F f " PG 7 F m G &-21
a 0f;/ 1 O')(87! Y D FOf " PG F f " PG ')(:7! F f " G 3 ( 7! ' & m
/ F G &-21
-;&-21 O0')f (8 7! Y D FOf " G F f " G ')(:7X! F f " G 3 ( 7X! ' m &-21
-;&-21 0')f;/ (:7X ! F G Y D F f " G FOf " PG O')(:7X! F f " G
2
0f;/ H ¤
2
0f;/ H ¤
. k - + a & a 1 0f / O'65_7X! Y D m 3 a 1 F
F f " G9H ¤
F f " G9H ¤
F f " G9H ¤
F f " G9H ¤
/ F Om G a 1
m m a 1 F G -;&-21 F G m G -;&-21 F G9H 5L
#
" "! #$ " #$ $!
O'65 7!
m m D a 1 F G -;&-21 F G m m ¤ a 1 F G -I&-21 F GIH / 8F Om G a 1
a . 8. k - + & a ; f / '65_1 7 ! m m Y D a 1 F G -I&-21 F G m m 3 a 1 F G -;&-21 F G9H '65_7! m m G D a F G F 1 m -;& -21
m ¤ a 1 F G -I&-21 F GIH m m 0 I f & 9. EDM &-21 A F
fI& G9H &-21
& 2 1 # ' " O')(87! £ '8>?7A@ B m ! / fI H = &-21 D m m 10. ED &-21 A F
0 I f & 3 &-21
&-21 fI& G9H '#" ! O ) ' 8 ( 7 ! m £ '8>?7A@ B / fI H = &-2 1 D m m
11. D &-21 F
&m -2 1 fI& m 3 &-21 fI& G9H # ' " ' (87! m ! £ fI / £ fI &-21 D / O')(87! £ fI / £ fI H ¤ / 3 m m
12. D &-21 F
&m -2 1 fI& m &-21 fI& G9H ] ' " O')(:7X! m ! £ fI / £ fI &-21 D / O')(87! £ fI / £ fI H ¤ / m ']" . f < 3 f I +
13. k &-21 / / & ! & <1 O')(:7X! O')(:7X! <1
£ £ Y D H
&-21 f;/ <3 &-21 f;/ '* >? = 7R@CB 3
5P
+ (
!<
m ']" . 0 f f I +
14. k &-21 / / & ! & <1 O')(:7X! O')(87! <1
£ £ Y D H
&-21 f / & -21 f / '* >? = 7R@CB 15. 0f;/ F m G I+ -; . 2D ' 0f / 3 ' 0f / H ¤ 2 2 16.
17.
18.
19.
20.
21.
22.
0f;/ F m G I+ -; . 2D ' f;/ ' 0f;/ H 2 2 . + a 1 a 1 0f;/ 43 ¢ ' & f &-21 - / A D f " O'65 ! H . I+ a 1 a 1 0f;/ 43 ¢ a 1 '% & f &-21 - / A D f " O'65 ! H ( 7! ' m a . a k - + & a 1 f;/ F O G 1
O'65_7X! m m Y D a 1 F G 3 a 1 F G9H '65_7X! m m G F 3 £ F a a 1
1 G ( 7X! ' m a . a k - + & a 1 f;/ F G 1
'65_7! m m Y D a 1 F G 3 a 1 F GIH '65_7X! m m £ G F F a a 1 G 1
. k &-21 &-21 f / &-21 0f / 3 ¢ f &-21
I+ &-!& ')(:7! 7X! £ f;/ <3 O')(: £ Y D
&-21
&-21 f;/ H k &-21 &-21 0f;/ 3 &-21 0f;/ 43 ¢ / f &-21
I+ &!- & . ')(:7! 7X! £ f;/ O')(: £ Y D
&-21
&-21 f;/ H ;9N
¤
¤
¤
¤
¤
¤
¤
#
23.
24.
" "! #$ " #$ $!
k &-21 & -21 0f;/ &-21 f;/ f;/ 4 '#" 3 &-21 0f;/ O')(:
& 2 1 X 7 ! m . £ f /
I+ &-!& D
& 2 1 O')8 ( 7! £ ;f / H ¤ 3 & -21
k &-21 &-21 0f;/ &-21 f;/
']" &-21 0f;/ ') (: &7-2! 1 0f;/ 4 m . £ f;/
+ &!- & D
& 2 1 O')(:7X! £ ¤
&-21 f / H
1.17.2. Derivatives with respect to the order
M ! ' % & -21 M ! ' % & -21 M ! M ! M ]! ' % & -21 M ]! ' % & -21
1.
2.
3.
4.
5.
6.
3
0& 3 ""'
L $9J % ')( $ ! D
"" '
& <3
L 9$ J % ')( $ ! D
& ] $ (* '!
& ] $ (* '!
& <3
& <3
&
3 & <3 & ¤ & <3 & ¤ & "" ' $ ( '2! L $9J % ')( $ ! D 3
""'
# $ * ( '!
0&
# $ (* '!
$ (* '!
& H ¤
& H ¤
& H ¤
& H ¤
&
$ (* '! L $9J % ')( $ ! D ;
$ * ( '!
+ (
M ! a 1
7 a 0& C £ & 3h
1
7 3 a 1 & V 1 N
!< ,
7.
7#7 ( #$ N ! 1 &
#$ ( 1
& ! & 1 &
( #$ 1 ¢ £ ! 3 / -;&-21 & -;&-21 0& 1 N 1 & 1 ( #$ & ! ¢ 43 & -;&-21 0& 3 -;&-21 0& 1 N & 1 ""'
')( $ ! ' % & -21 L ')( $ ! J ')( ! a a 1 0& 1 & H $9% $ D " ' %
L -21 ')J ( $ ! % $ % &- a 1 0& 3 $ (* (:7X! 1 £ & $ (* Q(:7X! £ Y D -21 $ (* (:7! -21 & H £ & &- a 1 & &D
2 1 $ (* (:7! 3 £ & H
2 1 $ (* Q(87! £ 3 43 ¢ a a -;&-21 0& D & 1 $ (* Q(:7X! £ & H 1 - Q(:7X! $ (* a a ¢ £ 3 3 -;&-21 & &D & 1 $ (* (:7! £ @ B 3 & H = 1 M ! M ! 8. a a 1
1
( #$ 7 1 a & # N ! 1 & & 1 & 1
7 #7 ( #$ 7 £ ! a 1 & C & 3h N & 1
; ,
#
" "! #$ " #$ $!
¢ £ 3 / ;- &-21 & <3 ;- &-21 0& 1 N ¢
3
#$ 1! ( 1 & 1 ( #$ & N ! 1 &
')( $ ! a & H 1
& ;- &-21 0& -;&-21 & 0 1 ""' ' % & -21 L ')( $ ! 3 $&J % ' ( $ ! D a & <3 1 " ' %
L -21 ')J ( $ ! % $ % &- a 1 0& $ ( Q(87! 1 $ ( (:7! £ £ & H Y D & < 3
-21 $ (* Q(:7X! -21 £ 3 &- a 1 0& D
-21 & $ (* (:7! £ & H
2 1 $ (* Q(:7X! £ & 3 3 ¢ a a -;&-21 & &D 1 $ (* Q(:7X! £ 3 & H $ (*1 Q- (87! ¢ £ a a 43 -;&-21 & &D & 1 $ (* (:7X! £ @ @ B & H = 1 M ! 9. 1 -; 7 #7 ( #$ 7 £
! C
& <3h N & 1 -; & 1 ] #$ 7 1! ( 3 1 -; 0& V 1 N 1 & & &
#$ 1! ( £ ¢ 3 / &-21 & &-21 0& 1 N 1 & 1 ( #$ & ! ¢ 3 & &-21 0& 3 &-21 0& 1 N & 1 "" '
' % & -21 ( L ')( $ ! ')( $ ! 3 $9J % ')
( $ ! D 1 - & 1 - 0& H
;93
+ (
Y D
!< ,
" ' %
L -21 ')J ( $ ! % $ % 3 ¢ a &- -21 0& $ (* Q(:7X! 1 $ ( (:7! £ £ & & H 1 1 $ (* (:7! £ ¢ & 3 a &- -21 & &D 1 $ (* Q(:7X! £ 3 & H (* 1 Q (8- 7! $ ¢ 3 43 -; a 1 & &D £ &
2 1 $ (* (:7! £ & H
2 1 $ (* Q(:7X! 3 43 ¢ -; a 1 0& D £ &
2 1 $ (* Q(:7X! £ @ @ B 3 -21 & H =
#$ 1! ( 0& V 1 N 10.
1 1 -; & & & 1 -; 7# 7 ( #$ 7 £ ! 1 -; & C & 3 N & 1
( #$ 1 3 3 ¢ / £ &-21 0& 3 &-21 0& 1 N ! 1 1 & ( #$ & ! ¢ 3 & &-21 0& &-21 0& 1 N & 1 ""'
' % & -21 ( L ')( $ ! ')( $ ! J ')( ! & <3 0 & H 1 1 $9% $ D " ' %
L ')J ( $ ! % $ -2 1 % 3 ¢ a &- -21 0& 3 $ ( Q(87! 1 $ ( (:7! £ £ Y D & <3 & H 1 1 $ (* Q(:7X! £ 3 43 ¢ a &- -21 0& D & 1 $ (* (:7! £ & H 1 $ (* Q(:7X! 3 3 ¢ -; a 1 & D £ &
2 1 $ (* Q(:7X! 3 -21 0& H ;!5
M ]!
7
#
¢
" "! #$ " #$ $!
$ ( Q(87!
43 ;- a 1 & &D $ (* Q(:7X! 2- 1
£ &
-21 £ & H = @
@ B
M ]! & -21 0& 3 &-21 & £ &-21
3 & &-21 & 43 ¢ C £ & 1 -; 0& 3 3 ¢ 1 -; & ( #$ 1 3 &-21 0& 43 ¢ 1 -; & 0 1 N ! 1 & & &
¢ &-21 0& &-21 0& 43 1 -; 0& #$ 1! ( ¢ 3?43 1 -; & 0 1 N 1 1 &
3 7 C &-21 & <3 £ / £ & &-21 & <3 &-21 & ( #$ & ¢ ¢ ! 3 1 -; & 3 1 -; 0& 1 N & 1 7#7 ( #$ ¢ ! 7 &-21 0& 3 43 1 -; & 0 N & 1 ""'
' % & -21 L ')( $ ! ')( $ ! J ' ( ! -21 & -21 & H $&% $ D " ( L ' % -21 J ')( $ ! % $ % &- -21 0& 43 ¢ a -; a 1 & 3 $ 1
5 P5 ! 5 P5 ! £ & <3 $ Y D £ & H
2 1 2 1 ¢ 3 a -; a 1 0& 3 &- -21 0&
5 P5 ! 5 P5 ! £ & $
£ & H Y D $
-21 -21 @CB = M ! 3
12. &-21 & &-21
£ ¢ ¢ C & 0 &-21 0& 43 1 -; 0& 43 1 -; 0& ; ;
11.
+ (
3
¢ &-21 0& 43 3 &-21 0& 3
L1 1! (
1 & & &
#$
1 -; & <3 &-21 & 0 1 N ¢
0 & 4 3
&-21 1 -; 0& #$ 1! ( ¢ 43 1 -; & 0 1 N 1 & 1
¢ & &-21 & &-21 & 3 3 1 -; 0& #$ & ! ( ¢ 43 1 -; & 0 1 N & 1
7#7 ( #$ ¢ 3 7 43 1 -; & &-21 & 0 N ! 1 & ""'
' % & -21 L ')( $ ! ')( $ ! 3 $&J % ' ( $ ! D -21 & <3 -21 & H " ' % L -21 ')J ( $ ! % $ % 3 1 Y 43 ¢ & 3 ¢ -; a 1 &
& 2 1 5 5 ! 5 5 ! £ & $ Y D $ £ & H
2 1 2 1 ¢ ¢ M3 &- -21 0& 3 43 -; a 1 0& $
5 P5 ! 5 P5 ! £ & <3 $
£ Y D -21
-21 & H @CB = M ]! M ]! 43 ¢ a 1
¢ a 1 ¤ 13. 0 & ! 3
& 2 1 &-21
M ! 1 -; ¢ a M ! ¤ 43 1 &-21 0& 43 ¢ 14. 1 -; &-21
1.18. The Legendre Polynomials Pn (z) 1.18.1. Derivatives with respect to the argument 1. 2.
j S 0 fI& £ 3 ¢ f S a ;- 1 fI& 5n' ! % m ¢ G 3h9f
O -; SE 1 a -; fI& F ( * ' ! % 1L' J
;9:
= > I' @CB = > 'IC@ B
& '
-21 ¢ h 3 f & &-21 j f;/ 0 7 43 ¢ F G f 63hf -;&-21 j F 7
m m 4. DM Q3hf &-21 j F
GIH 43 ¢ F 7 G f9 Q3hf9 O -;&-21 j F
m a ¢ a f / 0 5. 1 3hf9
& &-21 j 3 ¢ O'6 5_7 F 7 1 G f9 a 1 -;&-21 ¢ 3hf9 O& -;&-!&4 j F 7
m 5 m a . 6. D - + 1 Q3hfI &-21 j F m G9H F 7 G f
-;&-21 63hfI -21 j F " m
5 m 7. D fg3h& &-21 j F m G9H 43 ¢ F 7 G f fT3h& -21 j F " m
5 m S S 8. DM fg3h& &- -21 j S F m GIH ( % '2! % f + S -; . 0fg3h& - S -21 j S F 5 m G = > -; m SEa 1 . 0fg3h& SEa j S F 5 m GIH 9. DM - + m n 5 ' ! % 43 ¢ SEa a 1 . fg3h& S j SEa F 5 m % f
+ m 5 m ¤ j f
10. D 0Q3hfI 2j F m GIH D F " m G9H E( m S 11. D 0 3hfI& j S F ( m GIH " ' O ( m !% C7 ( ( ! ! (* > S . ' '! % 0 3hfI& + -; j S -; F ( m G = m (8 S S 12. D &- -21 0fg3h& j S F m ( m GIH " ' ' ! % m S- -21 fg3h& + S -; . j S F m (8( G = > ( O ' ! % m m -; 1 ( L ' J
. E S a j
+
S 13. ED 0 3hf F ( m GIH 1 43 ¢ 5n% '! % 3hf9
- + SEa a 1 . j SEa F ( m 3.
; <
G ¤ G ¤ G ¤ G ¤ G ¤ 'I@ B G ¤
'I@CB 'I@CB
G ¤
+ (
14.
15.
m
ED 0fg3h& -21 j F#" ¢ 3 GIH F m
a 1 D a 1 j a 1 F " ¢ 3 GIH 3 ¢ 'g 5 7 F
L1
7 G 0 fg3h& -;&-21 j ( m, ¤ ) * 7 G 63hf9 -;&-21 j E( m-, ¤ )+*
k &-21 0fI)3 ¢ -21 j / ¢ 3 fI F 7 G ;- &-21 fI 3 ¢ -I&-21 j F 7E( 7 G ¤ m
7 ¢ 17. D &-21 3hfI& j
F 7( m GIH 3 ¢ F 7 G -;&-21 j / ¢ 3hfI ¤
S 18. ED 0 3hf j S F ( m GIH
( ! O% '! % 3hf9
S -; j S > 'I@ B F
- ( m G = a S 19. 1;D 0 3hf9 O j S:F ( m GIH
( O')! % (:7X! % 3hf S -;&-21 j S > 'g5 7R@CB F ( m G =
- &-21 m 20. ED -;&-21 0f9 3h
j F (8 GIH m
7 43V F G f - &-21 f9 3h
j F m G ¤ m (8 m S S 21. D &- -21 0f 3h j S F m (8 GIH
m (*! % '! % f9 - S -21 0f9 3h
S -; j S > 'I@CB F m (8 G =
- m a S S 22. 1;D &- 0f9 3h O j S:F m (8 GIH
! % m 3 (* a S ')(:7X! % f9 1 - -21 0f9 3h
S -;&-21 j S - &-21 F m (8 G = > '65_7R@CB ¢ ¢ a S 23. &-21 3 f9
& 1 < 3hf9
& &- -21 j S 0f;/ ! (* % '! % F m G -21 ¢ 3hf9
& - S -21 j S > 'I@CB
- 0f;/ =
16.
;9L
24.
25.
26.
27.
) ' * '
m ED 0Q3hf9
a 1 ED S 0 Q3hf9
&- S -21 j S:F GIHOH m (*! % '! % F m G S ;- 63hf9
- S -21 j S > 'I@CB
- F G = m 5n D P D fT3h& S j S F m (* G9HOH D (*% '! % H fg3h& S -; j S F m 5(8 G = > 'I@CB -; m m 5 D P D SEa fg3h& - S -21Oj S F m (8 GIHOH D 5n% '! % H f S fg3h& - S -;&-21 j SEa F m 5(8 G ¤ m m 5 ED PED fT3h& - S -21 j S F m (8 GIHOH ¤ D 5:% '! % H fg3h& - S -;&-21Oj SEa F m 58 ( m G
1.19. The Chebyshev Polynomials Tn (z) and Un (z) 1.19.1. Derivatives with respect to the argument 1. 2. 3.
4.
5.
6.
ST fI& 0 £ &-21 3 ¢ f S m 5 EDM -1 ] fg3h& F m (8 GIH 3 m 5 S
0f9 3h
S ) m (8 , ( ! O% '! % f9
5 S S
&- -21 0 3hf9
S ) ( Y - S -21
> '8>?7A@CB 0 I f & = -; 7 ¢ F G -;&-21 0fg3h& ¤
3h
S -; m m , 3 f9
S -;
m 5 S -; ) m (8 , ! % ( O'! % f9 5 m S -; ) ( m ,
= > 'I@ B = > 'I@CB
m
D0 f 3h S S F m (8 GIH m (*! % '! % 0f 3h S -; S > I' @ B F (8 G = m
- m
a 1;D0 f9 3h
S S:F m (8 GIH ! % m (* ' 5 R7 @CB O')(:7X! % ; f9 3h
S -;&-21 S - &- F m (* G = > g ;9P
+ (
m
ED0 f9 3h
SEa 1 SEa 1 F m (8 GIH m (* 5_'g75 ! % 7! % f9 3h O S ;- a 1 S m (8 a F
- 1 S 8. ED0 3hf9 O - S:F ( m G9H ')(:7X! % . 3 5n(: 7X! % 3hf9
- + SEa SEa F ( m G S S 9. EDM &- -21 3hf9
S F ( m GIH
(*! O% '! % f9 - S -21 3 f9
S -; S ( m F
- a S S 10. 1;D &- 0 3hf9 O S F ( m GIH
! % 3 (* ' (87! % f9 a
- S -21 3hf9
S -;&-21 S - &- F
P1
7.
= > 'I@CB G
= 5n'*>?7R@CB G
= > 'I@CB
( m G ' 5_7R@CB = > 6
ED &- S - 0 3hf9
SEa 1 SEa 1 F ( m GIH ( O5_'67X5_! % 7X! % f9 - S - 0 3hf9
S -; a 1 S a F ( m G
- 1 = > 'I@ B m 5 S 12. ED &-21 PED fT3h& S:F m (8 GIHOH £ - (*! O% '! % -21 ] fg3h& S -; S F m 5n * ( G = > 'I@ B m -; m 5 S a S 13. D 1 D &- -21 fg3 S F m (8 GIHOH (*! % '! % F m G - S -21 fg3h& S -; S F m 5(8 G = >_'I@CB -; m m 5 S a 14. ED 1 PED -21 0fg3h& - S8F m (8 GIHOH £ - £ fT3h - S -; SEa F m 5(8 G ¤
m SEa &-21 fT3h& - S S F m 5(8 GIHOH 15. ED &-21 PED m £ - £ f S -21 0fg3h& - S -; SEa F m 5(8 G ¤
m £ fI a 1 0fI& = > 'I@CB 16. ST0fI& S -; 11.
: N
17.
18.
5 m n S ; f9 3h
S ) m (* , 5n ! % (* '65n ! % ;0f9
m S S &- - f9 3h
S ) m 5n O! % (* '65n O! % f - S - 0f
T n (z )
3h
S ;- S ;- 5n (* , 3h S -; S -;
U n (z )
m 5 ) m (8 ,
= > 'I@ B
m 5 ) m (8 ,
= > 'I@CB
m
D ;0f h 3 S S F m (8 9G H m (* O5_'675 ! % 7! % ; f9 3h O S -; S F
- m (* G SEa 1 SEa F m (* G9H 20. ED ;0f9 3h
1 m 5: O! % m (* O'65n ! % ; f9 3h
S -; a 1 S - a 1 F m (8 G S 21. D 0 3hf - -21 S F ( m GIH 43 ¢ 5n% '! % 3hf9
- + SEa . -21 SEa F S S 22. ED &- - 0 3hf9
S:F ( m G9H
( O5_'67X5_! % 7X! % f9 - S - 3hf9
S -; S
- F ( m G S - & 0 3_f SEa 1 SEa F ( GIH 23. D &- ! m
1 ( O58' O58! % O! % f9 - S !- & 0 3_f9
S -; a 1 S a F (
- 1 19.
24.
25.
26.
m 5 D & -21 P D &- S 2- 1 fg3h& S S F m (8 G9H H m S m 5n O! % * S ( 'g5: O! % F G - -!&4 0fg3h& -; S -; F m m 5 D a 1 D 0fg3h& S S F m (8 GIHOH 5: O! % m £ - (* O'65n ! % 1 fT3h& S -; S -; F m m 5 D a 1 D SEa a 1 0fg3h& - S - S F m (8 GIHOH 5: O5_'67X5_! % 7X! % F m G SEa 1 fg3h& - S -;&
:1
= > 'I@ B = > 'I@CB ( m G ¤ = > 'I@CB m G = >:'I@ B
5 (8 G
= > 'I@CB
5n (* G
= > 'I@ B
m 5 ¤
ES a F m 8 ( G
+ (
27.
,9N1
m 5 ED &-21 PED 1 0fg3h& - S - : S F m (8 GIHOH ' 5 7! % ¤ £ - 5n 5_g S -;&- ES a F m 58 7 ! ( G % g f h 3 & m
1.20. The Hermite Polynomials Hn (z) 1.20.1. Derivatives with respect to the argument
% £ * ( '! % fI S -; 0fI& = > 'I@CB 1. ST fI& 0 2. -21
7
ST0f / 4 3 ¢ SEa £ S _ F 3 G -;&-21 S-;&-21 0f9
& ¤ 3. SEa 0f;/
1 3 ¢ SEa £ SEa 1 ? F3 7 3 G fI -; a 1 -; a 1 0f & ¤ S ! % S S T f;/ 0 (* '! % - S -21 S - 0f;/ = > 'I@CB 4. &- -21 S 5. &- -!&4
SEa 1 f / 0 5_7X! % (* '65_7! % - S -!&4 S - a 1 f;/ = > 'I@ B m % m ( '2! % 43 £ fI -;&-21 S -; F G = > 'I@CB 6. &-21 S F G m S S:F m GIH 43 ¢ ( ! O% '! % S -; S = > 'I@CB 7. EDM
- F G SEa 1 SEa F m GIH 8. DM
1 m 3 ¢ (* 5_'67X5_! % 7X! % S -; a 1 S > 'I@ B a F
- 1 G = 7 m m ¤ ¢ £ _ S S 9. DM &-21
F 3 G -21 S-;&-21 ) ,
S F GIH 3
m 10. ED &-21
SEa 1 F G9H 3 ¢ S £ SEa 1 _ F]3 7 3 G fI -!&4 -; a 1 m ¤ S ) , W W W W 11. ED - S 0fI& H 3PfI - SEa fI& ¤ W W W W m m S F G9H f -;&-21 - SEa F G ¤ 12. D &-21 - :,
13.
14.
15. 16. 17.
18.
19.
20.
W
# . '/
ST0f;/ H 3 ¢ S £ S -I&-21 - W -;&-21 f9
& ¤ ES a W ED V- SEa 1 0f;/ H 3 ¢ S £ SEa 1 fI -I a 1 #- W -; a 1 0f & ¤ SEa ' W W ( X 7 ! D SEa &-21 - ST f / H ' S -21 - SEa 0f / ¤ W W ( 7X! ' S E S a ' V- SEa a 0f;/ ¤ D V- SEa 1 0f;/ H
1 W m D &-21 - S F GIH 3 ¢ SEa £ S -1 - W -;&-21 m ¤ SEa ) , W m ED &-21 - SEa 1 F GIH 43 ¢ SEa £ SEa 1 fI - &4 - W -; a 1 m ¤ SEa ) , W m S D - -21 - S F G9H 7 ' - S -;&-21 - W SEa F m G ¤
W m ED - S -21 - SEa 1 F G9H 7 ' - S -;&-21 #- W SEa a F m G ¤
1 ED -21 -
1.21. The Laguerre Polynomials Lλn (z) 1.21.1. Derivatives with respect to the argument 1. 2. 3. 4. 5. 6.
(Q(*')(:7 '65_7 c & c 0& c & <3 c a 1 0& ¤ Sc 0fI& 3PfI Sc a -; fI& c Sc fI& 0 3 ¢ 3 d 3 c -; &- S -21 Sc 0fI& 43Pd 3 - S -21 m m D &-21 S F GIH f -I&-21 S -; F G : 3
¤ c a 1 0&
= @ B = > 'I@CB
¤ Sc -; 0fI& Sc -; fI&
= > I' @CB = > 'IC@ B
+ (
m m EDM &- c -21 Sc F GIH 3Pd 3 - c -21 Sc -; F G ¤ m ¢ m S 8. DM Sc F GIH 4 3 43Pd 3 S -; Sc -; F G 43fI - c a fI& ¤ 9. - Sc 0fI& S n 5 ' ! % c -;a fI& ¤ 10. c - Sc 0fI& % c -; - SE a2SEa - Sc fI& 0 5n% '! % c a2S - SEc a fI& ¤ 11. c m m ¤ a 12. &-21 - R Sc 0 f - R Sc F G 5:'! % m ¢ G 3 % - c -21 -R SEc -;a 13. &- c -21 -R Sc F m S 14. - c - -21 #-R Sc F G 3 ¢ 5n% '! % - c - S -;&-21 -R
, ,
7.
= > 'I@CB
m F G ¤ m ¤ SEc a F G
1.21.2. Derivatives with respect to the parameter 1.
-/' M ! & -21 7 ')( $
= @ B
c &
1.22. The Gegenbauer Polynomials Cnλ (z) 1.22.1. Derivatives with respect to the argument 1. 2. 3. 4. 5. 6.
7.
S c 0fI& £ fI 0d2 S c a -; 0fI& = > 'I@CB &c a2SEa &-21 c S 0f;/ 0d2 &c a2S -21 c S a f;/ ¤ c a2SEa &-21 c SEa 1 f / 0 d2 c a2S -21 c SEa a 1 f / ¤ &- S -21 c S 0f / 0d2 - S -21 c S a - 0f / = > 'I@CB &- S !- &4 c SEa 1 0f;/ 0d2 - S -!&4 c S a - a 1 f;/ = > 'I@CB 5n'! % 5n O')(:7X! % % (87! % % SEa &-21 ¢ 3hf & 4c -21 c S 0f;/ £ -; % 7 Y C7 ( &! ' S -21 ¢ 3hf9
& c -;&-21 c SE-;a f / ¤ n 5 ' ! n 5 O 6 ' _ 5 7X! % % % 5 7! % % SEa ¢ 3hf9
& c -21 c SEa 1 0f;/ £ -; % 7 Y C7 ( &! ' S ¢ 3hf9
& c -;&-21 c SE-;a a f;/ ¤ 1 :95
0 ' /
£ fI d2 -;&-21 c a F m G = > 'I@CB S -; m S c F ¢ 3 G9H 9. (* 6( ! ' m C7 ( &! ' &- c fT3 £ & c -;&-21 ES c -;a F ¢ 3 G ¤ 5 m 5 £ m n ( G
d2 f 0fg3h& -;&-21 S c a -; F m * 10. D 0fg3h& &-21 S c F m (8 GIH = > 'I@ B m 5 11. D &c -21 fg3h& &- c S c F m (8 GIH &! ( ( 43 ¢ 5n% '! % Q(*J O 1 '! ( L ' c -;&-21 fT3h& &- .c c -; F m 5(8 G ¤ &(' SEa m m 3 ¢ S d2 S -; c a F m G S = > 'I@CB 12. D c S F GIH
S - SEa 1 c SEa F m GIH 3 ¢ 0d2 S -; a 1 c S a a F m G 13. ED
1
- 1 = > 'I@ B m S 14. D - c - 0Q3hf 4c -21 c S F GIH
7 3 £ -I 5n% '! % 5n O(8')7(:! % %7X! % % C7 ( &! ' - S - c Y 0Q3hf c -;&-21 c SE-;a F m G ¤
m S 15. ED - c - -21 0Q3hf9
c -21 c SEa F G9H
1 43 £ -; 5n% '! % 5n 5_'67X5_! % %7! % % C7 (7 9! - c - S -21
' Y 0f9 3h& c -;&-21 c SE-;a a F m G ¤
1 5 m S S 16. D 0fg3h& &- -21 S c F m GIH ! ( C7 (* Q( ! ' 5 m . C( 7 9( > 'I@CB Q( ! ' (9! ( "' f
I+ S -; fg3h& - S -21 S c F -; m G = a2SEa &-21 S c F 5 m GIH S 17. ED - c - 0fg3h& c m n 5 ' ! % 43 ¢ SEa . fg3h& c a2S -21 SEc a F 5 m G ¤ %
+ f c m S 18. ED 0 3hf9
- c - S c F ( m GIH 3 ¢ 5n% '! % 3hf9
- c - + SEa . c F G ¤ SEa ( m 8.
m EDM &-21 S c F IG H 43 DM &- cI fg3 £ & 4c -21
£ -; 5n% '! % C7
:;
+ (
, , ,
1.22.2. Derivatives with respect to the parameter 1.
2.
' / M ! 5n')(* $ d2 3 d2 c & $ ( 5n')( $ ! &c - & 1 = = @ O!C@CB 7 D F d 7 G 3 Fd G 3 £ £ d2 £ £ d £ H c & '#")" $ 5 9! & -21 7,5h C( 7X! £ ')( $ ! P5 $ 5:'! c & = @CB
3.
4.
£ £ d2 <3 £ d2 c & 7 (&! " 3 P5n'! " F $ $ 5: Q(:7G 1 7 5n')(* $
£ 3 ¢ c a & &-
&- &
c
= = @ O !C@CB
= = @& C! @MB
1.23. The Jacobi Polynomials Pn(ρ, σ) (z) 1.23.1. Derivatives with respect to the argument 1. 2. 3. 4. 5. 6.
7.
8.
5 65n'65_7 . ¤ a . 7<5 j + 1Rl & <3 j + l & 5n')(*'2 . . + -21Rl 0& 7(8 j + l & ¤ 7 . a . ¤ j + l & <3 j + l 1 & '65n'2 # 7 6(85n ' ! j + l V-21 . & <3 65n j + l . 0& 7 8 ( . m . ¢ j S + l fI& F G j S + -;a 9l a 0fI& = . M ¢ fI& j S + l 0fI& 3fI 3_3 ¢ fI V-; j + a 9l -; S . ¢ S M fI& &- -21 j S + l 0fI& 43fI 4383 ¢ fI - S -21 j + a 9l . 0fI& = S -; . ¢ fI& a a2SEa j S + l fI& f ¢ ¢ fI& a a2S j + a 9l S
. j + l & 5n'! 3 ]7 * ( j 5 g5n'65
7(*
: :
= @ B
.
> 'I@CB 0fI& ¤
.
> 'I@ B 0fI& ¤
&' 3 465 78
. M ¢ 3hfI& ¢ If & j S + l5n 0fI& 43 £ If % '! % ¢ 3hfI& -; ¢ fI& -; j + -;9lV-; . fI& ¤ SEa ¢ SEa a ¢ fI& j S + l . fI& 10. M 3hfI& 43 £ fI 5n% '! % ¢ 3hfI& a2S ¢ fI& -; j + lV-; . fI& ¤ SEa . ¢ a 11. M fIQ3 j + lC-;&-21 fI& m 5_7 ¤ 5_5 77! ! ' f fIQ3 ¢ a 1
, ' ) * . k a 1 j S + lM1 - S 5 -;7 ! ¢ ( fI(:
7X ! 12. ( (*'65_ 7! ( ' - a 1 j + -;9lM1 - SEa . ¢ fI
¤ S . m 13. ED &-21 j S + l F G9H - ¢ F3 m G -;&-21 j + a 9l a . F m G = > 'I@ B S -; . m 14. &- -21 $ f9 j S + l F G . m f 4 3 83 - -21 0 fI -;2j S + a 9l V-; F G ¤ . m S S 15. D 0 fI &- -21 j S + l F GIH f 3:3 S -; 0 fI - S -21Oj + a 9l . F m G = > 'I@ B S -; S fI a a2SEa j S + l . F m G9H 16. ED - - V- -21 0 . m 43fI ¢ - - V- S -;&-21 0 a a2S j S + a 9l F G ¤ I f . m a 17. ED - - &-21 Q3hfI 0 fI j S + l F GIH . m £ fI 5n% '! % - - a &-21 0Q3hfI -; $ fI V-; j SE+ a -; 9l V-; F G ¤ . m SEa a S 18. ED - - V- -21 0Q3hfI fI j S + l F GIH £ fI 5n% '! % - - V- S -21 0Q3hfI a2S fI V-; j + l -; . F m G ¤ SEa . ¢ m 19. ED &-21 j S + l F 3 G9H F m G ¢ -I&-21 j + a 9l a . F ¢ 3 m G = > 'I@ B S -; 9.
:<
+ (
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
,931
. m m . ED S j S + l F ¢ 3 G9H 43 ¢ 3 g3 S ;- j S + -;l a F ¢ 3 G = > 'I@ B m . ED - - V- S -21 j S + l F ¢ 3 G9H 43 ¢ ¢ - - V- S -;&-21 j + l a . F ¢ 3 m G ¤ S m . ED &- -21 j S + l F ¢ 3 GIH 3 ¢ 3 ¢ - -21 j + -;9l a . F ¢ 3 m G ¤ S m . ED S 0fg3 £ & j S + l F ¢ 3 G9H m . £ 3 83 S fg3 £ & -; j S + l V-; F ¢ 3 G ¤ m . ED S 0fg3 £ & &- S -21 j S + l F ¢ 3 G9H . m f 4 3 83 S -; 0fg3 £ & - S -21 j S + -;a 9l F ¢ 3 G = > 'I@ B m . ED S 0fg3 £ & &- S - -21 j S + l F ¢ 3 GIH m . 43 £ 3 g3 S 0fg3 £ & - - S -21 j S + -;9l F ¢ 3 G ¤ m . D - - V- S -21 0fg3 £ & j S + l F ¢ 3 GIH £ 5n% '! % - - - S -21 fT3 £ & V-; j + l -; . F ¢ 3 m G ¤ SEa m . D &- - V-21 fg3 £ & j S + l F ¢ 3 GIH £ 5n% '! % &- - -21 0fg3 £ & V-; j + -;9l -; . F ¢ 3 m G ¤ SEa m . ED - - V- S -21 0fg3 £ & a2SEa j S + l F ¢ 3 G9H £ 5n% '! % - - - S -21 fT3 £ & a2S j + -;9l . F ¢ 3 m G ¤ SEa . m 5n ED a 0fg3h& j + lC- -;&-21 F m (* G9H 5 5_77X! ! ' f D a 1 F " ¢ 3 G9H ¤ ' m . 5 m D - -;&-21 63hf9 2j + lC- -;&-21 F E( m GIH 3 ¢ 5_5 77! ! ' - -;&-21 D a 1 F " ¢ 3 m QG9H ¤ ' : L
K
E
D
1.23.2. Derivatives with respect to parameters
' 3 465 78 M ]! . £ ¢ <3 - ¢ 0 j + l 0& 5 g5n $ 5_7 g5 $ 5_7! ']")" . & -21 ')( $ !9 5 65 $ 5n'65_7X! 5 65 $ 5_7X! '#")" j + l & = @CB ' 3 465 78 M ]! . £ ¢ <3 ¢ 0 j + l & 2. 5 g5n $ 5_7 5 $ 5_7X! '#")" . & -21 ¢ &- ')( $ ! 5 g5 $ 5n'65_7! 5
g5 $ 5_7! ']")" j + l & = @CB 4 3 ' 3 465 78 M ! 5_7! ' 3. 7 . 5_7 7X! 43 ¢ a 1 ( 5: 'g(*5 '7! "! F $ 7 5 7 $ G F (: + a l & G j ' " &1 = = @ 7X!C@CB ' 3 465 78 M ! 5 g 5n'65_7! ' 4. g5 7! ' 5 g5n $ 5_7
5 g5_7! " .
5 g5_7X! ' &-21 ')( $ !# 5 g5 $ 5n'65_7X! 65_7! " j + l & @ = = !C@CB 1.
1.24. The Complete Elliptic Integrals K (z), E (z) and D (z) 1.24.1. Derivatives with respect to the argument 1. 2. 3.
4. 5.
7 ¢ 3hf9
& P K f / 0 F G 9f ¢ M 3hf9
& k &-21 K f / 43 ¢ k M ¢ 3hf & &-21 K f;/ 43 ¢ F 7 G
7 ¢ 3hf9
& &-21 < E f;/ 0 F]3 G k ¢ 3 f9
& &-21 P &-!&4 E f;/ 3 : P
K 0f
/ ¤
7 F G -;&-21
K f
/ ¤
f9 ¢ h 3 f9
& -;&-21 K f / ¤ 7 m ' F G 7 ( m E 0f;/ ¤ "' ¢ F]3 7 G 7K ( E 0f;/ ¤ m
+ (
, ;
k ¢ 3 f & -21 P ¢ 3 f & &7 -21 E f;/ 3 ¢ F G F G f9 ¢ 3 f9
& -I&-!&4 E f;/ ¢ ¢ 7. k 3hf9
& 9M 3hf9
& &-!&4 E 0f;/ 3 ¢ F]3 7 G F 7 G f9 ¢ 3hf9
& -;&-21 E f / ¢ ¢ F 7 G F G f E f;/ a 8. 3hf & 1 3hf & -1 E 0f;/ ¢ ¢ a 9. M 3hf9
& 1 k &-21 3hf9
& -1 E 0f / 3 ¢ F 7 G -;&-21 E 0f;/ 7 ¢ a 10. 1 3hf9
& D f;/ 0 F G F G f9 D f;/ ¤ 7 7 ¢ F3 G F G f9 D f / ¤ 11. &-21 3hf9 O& D f / 0 ¢ 12. k; 3hf9
& P &-21 D f;/ 43 ¢ F 3 7 G F 7 G f9 1 -; D f / ¢ a 13. k -1 3hf9
& P 1 D f;/ 3 ¢ F 7 G F G -;&-21 D 0f;/ ¢ a 14. k 1 M 3hf9
& &-21 D f / 3 ¢ F 7 G f ¢ 3hf & -I&-21 D f;/ 6.
¤ ¤ ¤ ¤
¤ ¤ ¤
1.25. The Legendre Function Pνµ (z) 1.25.1. Derivatives with respect to the argument 1.
2.
D j
D j
m ! ")" £ F m IG H 3 If 0f 3h -I $9% T3 ¢ 4383 0f9 3h O &--; £ m ! ")" F m GIH 3 fI 0f 3h -I $9% Y 0f9 3h O -; &-
Y
<9N
L ']")" ')( J $ 5 ! ']")" a + 1A- . F Ggj - F m m ( L'#")" J ')( $ (! '])" " a + a 1 . F Ggj a F m m
G ¤
G ¤
&
ED0 f9 3h O j F m IG H . 43 ¢ T 3 ¢ 43:3 ; 0f 3h + -; j -; ¢ 0f 3h - + a . j a F G ¤ 4. ED0 f 3h - j F m G9H 4 3 m a . 5. DM &- + -21 fT3h& j F#" m GIH £ -; f 43:3 ; - + a . -21 fg3h& + -; . j -; F#"
-; . a a 6. EDM + -21 0fg3h& - j F " m GIH 43 £ -; f
I+ a -21 . 0fg3h& - + a . j a F#" a . a 7. EDM + - -21 fT3h& j F#" m GIH £ -; f )3 ¢ I+ - -21 . fg3h& + -; . j -; F#"
a . a 8. EDM + - -21 fT3h& - j F " m GIH 43 £ -; f
I+ - . -21 0fg3h& - + a . j a F#" -; m 9. EDM 63hf9 j F " GIH 3 £ -; 3:3 ; I+ -; . Q3hfI + -; . j -; #F "
-; m Q3hfI - j F " QG9H 10. ED £ -; I+ -; . 63hfI - + a . j a F " -; m . a 11. ED - + 1 0Q3hfI j F" QG9H 3 £ -I T3 ¢ - + a a 1 . Q3hfI + -; . j -; F "
a m a . F " GIH 12. ED - + 1 0Q3hfI - j # £ -; - + a a 1 . 0Q3hfI - + a . j a F " a 3.
F m G ¤
¤ m G ¤ m G ¤ m G ¤ m G m ¤ G m ¤ QG m ¤ QG m ¤ G
1.25.2. Derivatives with respect to parameters 1.
j & 3
5_7 3
1
j & £ a 1 $ J 5n 1 L' " ! % $ ¢ 3 a 1 & = = @ #B !C@CB &<
+ (
, ; ,
j & -;&-21 3 j & ¤ j 3. & &-21
)(* '65_7 3 j G ¢ 3h
& 3 ¢ £ &-21 F )( O'6 5 $ 5_7 ' &" -21 £ 65n '65_7 Y $ L G G9H
3 D F F J
- a 1 L" J Y j -; 0& 43 £ & - ¢ 3h
-
l -21
5 Q(87! Y -21 % $ $ ( Q(:7X%! % £ & - ¢ 3h
j a -;&- & ¤ -21
7 7 D F 3:3 G 3 F 3 G 3 HTj 0& 4. &-21
']" ¢ 3h
1 ( L ' 1 5 L ' J
J
a ' ) * ( O 6 ' 5 &- -21 J - & L Y $ L 3P £ F G 63 3 % ')( $ ( ! J & 5 Q 8 ( 7 ! 7 Y j a & 43 £ & - ¢ 3h O - -21 % $ $ ( Q(:7X%! % F 3:3 G l -21
- Y £ & - ¢ 3h
j - a a & ¤ -21
3 ¤ j j 5. & -;&-21
0& &-21
6. j % & ! £ 5n ¢ £ C j 3& O! ] j 0& 3 £ £ 5_7X!A 5: O! 3 - 4 R0 j a 1 & j a 1 3P& ¢ 3h j 1 a 0& j 1 a 3& 0 ¤ 7 (8 " 7. D j 0& H DC3 £ C £ &Q3 ¢ 3h
1 1
_F ¢ 3h G F ¢ 3 7 (8 G F ¢ 3h G F ¢ 3 7(* G9H = C7 ]! @CB 2.
< ,
6 !
1.26. The Kummer Confluent Hypergeometric Function 1 F1 (a; b; z) 1.26.1. Derivatives with respect to the argument
m !' m ! m 5:' ! @ VB B 7 @MB 1. D N == ! ' 1 N 1 J 5n' L 1 1J LH m ! m 5n' @ VB B 7 7R@MB a 2. EDM &-21 N == 1 1 J L H 0fI -21 1 N 1 J ! L m ! m ! ¢ ¢ @ VB B 7X @MB 3. DM -21 N == 1 1 J L H 3 3O -;&-21 1 N 1 J (*' L m ! ( m ! ' m ! @ VB B 7 @MB H ! n 5 ' L 3 ' - 1 N 1 J == 4. ED
- 1N 1 J L m m (*' 5. DM -2 1 V
- 1 N 1 J ! L H 43 ¢ ¢ 3O -;&-21 #- 1 N 1 J (*!' L ¤ m ! m (*' a H K3hfI - -21 - 1 N 1 J L 6. EDM - &-21 - N 1 1J L ! @ B B 7 @CB == m ! m !' m 5n' ! ¤ ! 7. &-21 N 4 3 N ; & 2 1 ` ' 1 1^ 1 1 ^ 5n' ` m 5:' m! ¢ 43 0fI - -; 1 N 1 ¤ 8. - 1N 1^ ` ! m! ¢ m! ¤ 9. & 3 O - 1 N 1 ^ (* ' ` 1N 1^ ` m! m! ¤ ( ! m ! ' 10. &-21 #- C N ; & 2
V- C N 1 ` ' 1 1^ 1 1 ^ 5n ' ` m (*' m! ¢ 11. - V
3 3hfI - -; # - C 1 N 1 ¤ - 1 N 1 ^ ` ! g5 1 1 ( ' 6 ' 5
6 5 1 L' L' 1 a 12. N1 1 43V J J ! '%& 7 1 N 1 5n'6
5 ! ! = 7 ! = @ @CB & ( (*' '65 & W ! 7 " WU ! a !
1N 1 3# 1A- 1 N 1 5n' 13. ' ! ! = 7 ! = @ @CB <93
+ (
( (*'65 1 a
-21 1 N 1 14. ! 5 1 43 ¢ ¢ 3 £ O a - - &-21 N = * ( '
1 1 ! * ( ) ' ( a 1 15. - N 1 1 ! 43 ¢ £ 3 £ O a - - &- N ( ( 1 ! (* ' =
1 1
m m 5n'65 £ a a !
a 16. 0 I f 1 N 1 g5 1 1N 1 1
! =
m m 5:' £ a !
0fI 1A- 1 N 1 & ( ! 17. N 1 1 &
= W m 5n')( 1 a 18. - N 1 1 m! g5 ( 1! 3 ¢ £ J 1 L ' J ! 1 L ' - W N m 5n m '%& 7 1 1 m '6 5 = m 5n'6
5 ( & W a 19. - N m! 1 1 ( m ( & £ J & L ' J ! 1 L ' 1A- - W N m ' 1 1 m 5n' ! = '6
5 ( 1 W a 20. -21 - N m! 1 1 (')( 1 3 ¢ ¢ 3 £ fI a - &- -21 - W N
1 1 m (*' ! = '6
5 ( 1 W 43 ¢ £ 3 £ fI a a
- #- 1 N 1 21. m!
* ( ) ' ( W 1 Y - &- - V- N = * ( ) ' (
m 1 1 !
,9:1
7 @ @CB ! =
7 @ @CB ! =
7 @ @CB ! =
7 @ @CB ! =
7 @ @CB ! =
7 @ @CB ! =
7 @ @CB ! =
7
! =
@ @CB
6 !
m ! - 1N 1 1
m W 3# a F 7 3hf G - 1N 1 a m W a
- 1 N 1 !&
3V F 3hf G 1A- - W N m 1 1 &
22.
a
23.
W
( ' * g5 ! 1
7 @ @CB = ! =
(*')( (
7 @ @CB = ! =
!
1.26.2. Derivatives with respect to parameters
m ' 5! 7 L H SEa D1N 1Jg 5n% ' ' % ! %
1.
a 1 ¢ <3 ¢
S-; 3P&
;- C £ & <3 & -; 43& SEa SE a 7
3 $ SE;- a & - 43& £ 43 ¢ - -21 & 1 % ' % ! S " 3 5:'2! % -; J $ L $9%
( 7! Y C $ ¢ -; 43& <3 &-;- a 4 3& ¤ 1 m ! 1 N 1 1 (*' SEa
1
£ 43 ¢ _ a 1 ' $L F]3 7 G 3P& - J 1 ( 1 J S L J L ' 7 #7 ( ( 7X! 7 a Y % F 3 G S - 3P& N & ( $ ! (
7 ¢ £ -;&a -21 3P& G H 3 43 D F C SE S 7 7
$ -21 3 3 ¢ SEa $&% ( $ ! J LF 3 G -;&-21 4 3& ¤
2.
< ;
+ (
, <
3.
4.
5.
m ! 1 N 1 & (*'
&4
'& ( ]! ( & J L'& '65_7 D1N 1J m ! LH
7 #7 ( ' G 3& - N ! ( $ ¤ $ L F 3 J
7 1A- 43& fg3 ¢! & $ &-- a 1 43& --21 -21 0& ( ! " m 5 $ (:7 m 5 $ (:7 1 ¢ 3 0fg3 $9% m 5 $ (:7X! &- 3P& N J m 5 $ m 5 $ ( L ! m= 7 9BB B@CB ')( 5_7 m G 3 H D 1 N 1 J m ! 5 L H a . D £ ¢ <3 F +
& 1 7( (*' E- a 1 1- - ¢ a Y N '6 G 7( ' 1 1 5_ 7 ! 3 1 -; F ! 5 $ ( 7X! ' ' % ! " ' & -21 $ 5 1;- E- L 1 ¤ J $&% ! ')( $ ! 43& N a _ 5 7 $
1 1 E- 1 ! J L 7 ')( 5_7 m ( G 3 H D 1 N 1 J m ! 5 L H a . D £ ¢ 3 F + &- 1 a ( ' % "' Y N E- 'g5 1 7 !
1 1 E- a 1 L J $ 5 ; 1 E 3 L & 2 1 £ Y 1 J
f O ! ) ' ( ! & $ % $ ¡_ 3
& $ 5 1;- - ¤ Y N 1 1 $ 5_7 ! ( 1A-
6.
1.27. The Tricomi Confluent Hypergeometric Function Ψ(a; b; z) 1.27.1. Derivatives with respect to the argument 1. 2.
m m 5n' ED J ! L H 3 fI J 5n!' L 5 ' m m n ! L D a &-21 J ! L H fI 0g f 3 ¢ 2- 1 J <9:
= = @ VB B 7 7R@MB = = @ VB B 7 @MB
3. 4. 5.
m EDM -21 J ! L H 4 3 ¢ 0fg3 m ! ED # - J L H 43 V- m ! a EDM - &-21 - J L H 3
¢ -;&-21 m !(*' L J m ! 5n' L J ¢ - -21 - m (*' ! J
= = @ VB B 7X @MB @ VB B 7 M@ B ==
m 5:' m &-21 ^ ! ` fI -;&-21 ^ 5:'! ` ¤ m 7. - ^ ! ` 3 ¢ fI 0fg3 ¢ - -; m! m! ¤ ¢ 0 g f 3 8. & ` ^ ^ (* ' ` m m! 9. &-21
- C ^ ! ` - C -I&-21 ^ 5n ' m! m (*' - -; - ^ 10. - - ^ ` 6.
!
L
= = @ B VB 7 X@CB
m 5n' ! ¤ ^ `
` ¤ !
` ¤
1.27.2. Derivatives with respect to parameters 1.
m! ' LH D J 5n 3h O # 5n ! ' L &-21 J 7 & -21 ')(:7 3 ! KJ $ L 3 ¢ 5 $ ! 7 #7 &" D 5 $ (:7 N ( ! J m! D J 5_7 L H 3 - O 3 ¢
' (87 $ L O - - J % ! 5 $ ! 7 L 5 $ 1 N 1 5 $ 5_ J ( $
2.
3.
< <
+ (
K a2S 7 ES a 2- 1 0 & - a2S 2- 1 - 2- 1 3P& 3 1
Y
4.
, < ,
0& - 2a S - S 2- 1 43& = ' >?7R@CB m ( 7 ! ( 7 ' 7 ! ' G F3 ' 5 1 6 J $ L & ( ' L " S
J
Y - a2S -21 &
& <3
5.
7 #7 7 Y £ N ! 3 / <3 F 3 G 3 F 3 3 ]
&
a2S 7 3 a2- S -2- 1 & 1 Y / 1 - 43& <3 N 1 P5 ! 1 1 -21 1
J L 2 a S 7 7 5 ( ] ! $ ¢ L 43 F G F G S F ( $ 1 J J L 43 ¢ - -; D m ! L H a a J 1 Y 3 ¢ ¢ 0fI C 3 # "'& $ 5_7 $ 5 7 ! L C 7 ( ! _ 5 X 7 ! 3 N J $ ( m 5: $ 5n m "'& $
3 43fI 3 ¢ 9l - -21 43& ( 7X! " $ $&% J L 43fI - 3P& - 0 f
7 G £
7 3 G ¤
D
6.
Y
J
5 7
/ -21
!
( 7 ! % D C £ £ F GIH L H a &4
3P& a2S 7 - a2S -21 3P& / <3 - S -21 3P& a2 1 <9L
¢! 3&
' J $ L 1 - 0 & -21
¤
( ! % -21
Ψ(a; b; z )
' $ L 3P& J 5 P5 #$ a2S ( ! 1! ( 1 Y % 5_7! a2- S 43& N
P5
& P5 &
a2S ( 7X!
- S -21 43& ¤ 3 C V& - a2S -21 43& a2 1 ( 1L ( ( 1L ' 5 7 ' $ 5 !% J J H
D J L 0 7. ! % 5 1 L ' J $ L & (*' L " ! &4 -; J
a2S J
7 7 Y D £ F G 3 F 3 3 G9H J $ 5 ( L 7 Y / -21 - -21 43& / <3 - -21 43& 1 - 0& -21 1 ( % ( 7 7 1L' ' a2S $ 5 ( $ $ !% L L ! % J 5
1 L ' J & (*' L " J J J
5 1 5 1 ( #$ ( X 7 ! ! Y # 5 & 5 & J L 5_7X! N
3 / -21 ] C V& - -21 43&
-21 ( 7! ¢ ¤ Q( 43 / -21
P 3 & - -21 m! n 5 ' H L S 3? $ 0 J ! 5:' L 8. D J m " ( SEa &-21 5n')(:7 ¢ $ L 3 - (87! % J ( ]! " ( SEa &-21 5n')(:7 ¢ $ L 3 3P& 3 (:7X! % J a2S 7 Y & <3 & 0 - a2 S - S 43& <3 -21 - 0& a2- S - S 43& -21 -21 - -21 1 = >?7R@CB
<9P
+ (
9.
g ' 5 1- m! 3 D J m 5 L H a 0. O '65_
7! + 1A-
" " & -21 ' ( ! £ $ & & &-21 -; $&%
,9L1
3
£) 3 < 3
¤
1.28. The Whittaker Functions Mµ, ν (z) and Wµ, ν (z) 1.28.1. Derivatives with respect to the argument 1.
2.
3.
4.
5.
6.
7.
&- -21
A
l 0 fI & ¢ 7 F 3 G - -21 A 29l -21 A l 0fI& 43 ¢ 3 £ f
- + a 1 . A 2 l -;
- -21 A l 0fI& (g5 ¢ J 5_7! 1 L ' f
- - + a 1 . A ' 2 l a
-21 A 7 l 0fI& 3 ¢ F 3:3 G f
- + a 1 . A -; l 2 -21 A 7 l 0fI& 3 ¢ F 3 G f - + a 1 . A -; l 2
&- -21 A l 7 0fI& F 3:3 G F 7 3 G - -21 A -;9l -21 - A l 0fI& 3 ¢ f - + a 1 . #- A a l 2
0fI& ¤ 0fI& ¤ fI& ¤ fI& ¤ fI& ¤ 0fI& ¤ 0fI& ¤
1.29. The Gauss Hypergeometric Function 2 F1 (a, b; c; z) 1.29.1. Derivatives with respect to the argument 1. 2.
m !' !' m m 5n' 5:' ED N 1 J L H ! ' N 1 J 5n' L ! ! m m 5n' L EDM a &-21 N 1 J L H fI 2- 1 N 1 J ! ! L N
= = @ #B B @MB = = @ #B B 7R@MB
3. 4.
! $
m D -21 N 1 J L H 43 ¢ ¢ 3 -;&-21 N 1 J ! m ED0 ¢ 3h& a &-21 N 1 J L H ! 3 ¢ m ! ' ! ( ! ' ¢ h 3 & -21 N 1 J '
m (*' L
= = @ #B B X@ B
5 ' m n 5n' L
= = @ #B B @CB
!
!
m ED0 ¢ 3h& a - N 1 J L H ! m @ VB ( m ! ' ! ( ! ' ¢ h a
; n 5 3 & N '
1 J ' ! L == a m ¢ - N 1 LH 6. EDM -21 3h&
J ! (*' 3 ¢ ¢ 3 -;&-21 ¢ 3h& - N m (* @ VB
1 J ' ! L == ¢ a - N m L H 7. EDM -21 3h&
1 J ! ' (*' 43 ¢ ¢ 3 -;&-21 ¢ 3h& a - A-; N m (*(* ' L = = @ VB
1J ! m 8. EDM - a &-21 ¢ 3h& a - N 1 J L H ! 3hfI A- -21 ¢ 3h& a - -; N m (*' L = = @ VB
1 J ! m m 5n' : 5 ' ¤ m !' !' ¢ ! 9. &-21 N 4 3 N ; & 2 1 n 5 ' '
1 ! 1
1 ! 1 m m 5n' ¤ 10. - N 1 1 43 ¢ fI - -; N 1 ! 1 ! m m 11. &
- N 1 ¢ 3 - N 1 (*' 1 ¤ 1 ! ! m 12. - Q3 ¢ a &-21 N 1 ! 1 m ! ' ! ( ! ' - -; Q3 ¢ -21 N m 5:5n' ' '
1 ! 5.
L1
B @CB
B @CB
B C@ B
B @CB
1 ¤
+ (
13.
m N 1 ! 1
A- - &-21 Q3 ¢ a -
43 ¢ ( m ! ' ! ( ! ' ' a
m A- - -21 0Q3 ¢ a - A-; N 1 n 5 ' 1 ¤ !
¢ 3 - 0 Q3 ¢ - N m (*(*' ' ¤
1 ! 1 m &- - 63 ¢ a - N 1 ! 1 ¢ 3 &- - 0Q3 ¢ a - A-; N m (*(*' ' (*' ¤
1 ! 1 m - 63 ¢ a - N 1 ! 1 43 ¢ 3hfI - 0Q3 ¢ a - A-; N m (*' ¤
1 ! 1 ('65 1
a N 1 ! '6
5 g5 1 5n'6
5 43# F 7 G F 7 G !! ' & 7 N '& 7 1 5n'6
5 ! 7 @ & O!C@CB = ! = (')( g5 &
a N 1 ! '65 & 5:' 43V F G F 83 7 G !! '%& 7 1A- N '%& 7 5n '
1 ! 7 @ & 7X!C@CB = ! = (*')( g5 1 43 ¢ ¢ 3 £ a
A-21 N 1
a ! 5 1 7 @ !C@CB Y A- &- V-21 N ! =
1 (*' ! =
16.
17.
18.
19.
¢ - a N m 14. - 63
1 ! 1
15.
,9P1
L,
20.
(*')( a
A- N 1 !
1
! $
( 1 (*')( ! = 7 ! = @& O!C@CB 5n')( 1 7 7 a ¢ a 43# F G F G 21. 3h
&-21 N
1 ! ( ( !'& 1 Y ! '%& 7 ¢ 3h -;&- -21 N n 5 6 ' 5
7 1 ! 7 @ & O !C@CB = ! = 5:'g5 ( & a a ¢ a 22. ; 3h V-!&4 N
1 ! ( & 43# F G F 83 7 G ( ! ! ' 1A- ¢ 3h
-;&!- &4 N ' 5n'
1 ! = 7 ! = @& O!C@CB '65 ( 1 a a ¢ a 23. A-21 3h
- -21 N
1 ! (')( 1 (*' 3 ¢ ¢ 3 £ ¢ ( '
a - &- -21 3h
- -;&-21 N 1 ! = 7 ! = @& O!C@CB 'g5 1 a a ¢ 3 ¢ £ 3 £ a 24. A- 3h
- 1 N 1
a ! ( ' ( g5 1 (*')( Y - &- - ¢ 3h - A-;&- a 1 N (*')(
1 ! = 7 ! = @& 7X!C@CB m a 25.
N 1 1!
m 5n'65 5n'6
5 £ a 0fI O 51
a a N 1 !
= 7 = @& !C@CB ! 3 ¢ £ 3 £
a A- &- V- N 1
L 3
+ (
26.
27.
28.
m m 5n' 5n' £ a
N 1 &
fI O 1A- N 1 & ( !
! 7 @ & 7 = ! = m 7 ¢ a a
3h
-21 N 1 1 £ a F 3hf G a !
* ( ' (*' 7 m Y F 3 G a ¢ 3h a - &- V-21 N 1 1 5 ! 7 @ & = ! = m ¢ a a
3h
-21 N 1 & £ F 3hf G F 3 G !
m (*')( (*')( Y 1A- ¢ 3h O a - &- V!- &4 N
1 & ( !
7 @& = ! =
,9P1
!C@CB
!C@CB
O!C@CB
5 ( #$ m m ¢ a a a 1 29. 3h -21 N
1 1
! 5 5 ( #$ m m 1 3# fI ¢ 3hfT3 , ¢ a a 3h O -;& -21 N 1 5 1
g ! 7 @& O!C@CB = ! = m ¢ a a 3h
&-21 N 1 30. ; ( m ¢ Y 1A- 3h -;&- V-21 N
1
31.
¢ a a
3h N 1 43V a
m ( g5 1 #$ 3# 0fI ¢ 3hf-,
& !
( g5 #$ m 1 = 7 ! = @ !C@CB ( & !
7m 1!
7 '6
5 65_7 m F 3 f G a N 1 g5 1 ! 7 @& O!C@CB = ! = L95
! $
1.29.2. Derivatives with respect to parameters 1.
2.
3.
4.
5.
'65_7 m D N 1 J m 5 ! L H 5 7 ! " . a a O; ¢ 3h& -;&-21 ¢ h j + 1Rl -;&-21 ¢ 3 £ & 3 & 5 5 ! 9 $ % $ m &5 5 $ m 5 5 $ 5 $ 5_7 ¤ m Y N ! & J m 5 5 $ 5_7 m 5 5 $ 5 7 L '65_7 m ¢ 3h& - -21 m 5_7X! " 43& ¢ 3h& - D N H L P 3 I f ;
1J ! $&% 5 $ ! 5 $ 5_7 5 $ 5 $ ¤ 7<5 Y j + a 1Rl a -;&-21 . F 7 (8 G N m 5 $ 5 7 5 $ 5_! 7 ,-Q 1` & ^ & V7 1 N
1 ! 0 # M E7 (:7X! ¢ £ / 1 ¢ / ¢ 3 £ / ¢ 3 / <3 ¢ 3h& ¤ m D N 1 J 5_7 L H ! O - ¢ 3h& 1A- ¢ 3h& -21 ¢ 3hf O X 3hf ¢ < 3 ¢ 3hfI 7 7( m V7 ( 7 7( m V7 ( m V7 ( ¤ ¢ 7 ( ( 7 8 ( L 3 C7 ( m ! & N J ( m ( m 7 (8 L m 3h& N 1 J m ! ! mm 5 ! ( LH D N 1J m 5
a . ( 5_7 ')( 5_7 + &- 1 ') DC3 G GIH £ ¢ < 3 F 3 F a 5 a Y N E- '61 5_7 (E - 1
1 ! 7 ']" ' % C7,5 ! 3 " '#" 8 " ¢ ')( 5_7! 3 E- a 1 L 5 E- a 1 L J
J
7,5n')( 5n' 7 (*')( Y 3 F G O F G F G ¢ &
#$ a Y N 1 - 7 E ( - 1 E -
1 ! a1 ( (*' 7 1 (*') ( ')( 5_7 G F ' G G 4 3 O F F L;
+ (
a a 1 5 E- a 1 #$ Y N
5_7 1
1 ! a 1 $ 5 " $ 5 1 E - L Y &-21 J
$&% ! ')J ( $ ! $ 5 Y N
1
3 N1
4
-;
5 1;- E- L
1;- E- $ 5 5 ;1 - E-
$ 5_7 ! (
¤
1.30. The Generalized Hypergeometric Function p Fq ((ap ); (bq ); z) 1.30.1. Derivatives with respect to the argument 1.
m !95n' m ! m !' !' N ^ !95n' ! ` ¤ N ^ ! ! `
m ! ( N ^ ! ! ` 3 ¢ 3 -; 2a S N a2S m !! g g ^ = m ! N ^ ! ! ` ! ' ' ( % ! % m '#" a N a m !9!95n5n')')( ( ! ']" 1 1^ - -21 $ (*'65 5_7X! ' m ! " a $9% ! " -; =
2.
3.
5_7! ( ! 7! ` (*'65_ ')(:7 ')(* ' ( &B B B0@ B
'65_7 ! 7 ` ' ( 5_
!95n' '65 m ! 7 m !' m 1 £ !
N ^ ! `
F G ! ' a 1 N a 1 !95n'
1
! m ! a
1 N ^ ! ! ` !95:'g5 7 '65 ! m £ a 1 F G m ' & a N a !' & 1 1 !95:'g5 7 & !
4.
5.
')(:7' ( ')( 9BB B@CB
L :
#$
¤
&
#$
¤
1.30.2. Derivatives with respect to parameters
" m !" m m ! ( ! ¢ a 1 N ^ ! ` 3 $&% ! " ¤ ! (' m m ! a N a
1 ^ ! ! ` ( ' m m ! 3hfI 3 3hf a N ^ & ! ` ! " ' % '65 ( m !A ( m (*'! " & -21 ( 7X! $ (* $9% ')( $ !A $ ( m 5 ! 3 m ( 5 7! ' ')( $ ( $ m 5:' ( $ m !95 m ! Y & !95 & L P 3 & a a N ` ! 1 ^ J ! ( ' & m ! 0 a N a ( &' & ! m ! ` ! ` a N a ! O 3 1 1^ 1 1^ ! ' % & -21 ! " ( $ & m ! ')( $ ! a 1 N a 1 ! ` !' $9% ^ ! m ! m ( m a N
^ !! ` a
D F 3 G 3 F G9H a N m ! (*' 5:'
! !
" " 5n $ (*' a 1 L & -21 L £ J 5n &-21
' $&% ')( $ ! J a 5 $ (*' 1 L L J
J ! a (*' 5: $ ( ' #$ m 1 Y a N a a 5 $ (*
' ! & 1 1 !
(' m m ! a a N
1 ^ m 5 ! ! ` 0 f < 3 0fI f O < 3 f 0 (' m m ! Y a N a m 5 ! `
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1 ^ m 5 ! ! `
1.
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7.
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mm &N m 5
m ! 1
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1 O7 D ¢ h 3 &
3 K m 1 <7 5 7 (8 K *
3 N1 ,
7,5 <7 5 / 7,5 3 <7 5 H ¤ 7( 7(8 * 7 ( 7 8 ( ¤ £ K *
m m m ( N& ^ m 5_7 ! m ` 3 K F 5_7 5 7 G 1
7 Y _ K _ 5 7 5 , 5_75 5 75 7 ) 5_75_7 K F 5_ 7 5_7 G ¤
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m m & ! \¢ 3hfg )¢ ) 3hf 3f 3 K 3 h 3 f 3 3 ¤ Y & \ ¡ ) \ ¡ £ £ 3 3 f 3 f mm 5 m 5 ( m 5 5 1 10. N & m m (K5_7 ! 3 D F ,G F 3 3 ¢ G £ CH #! ( ( 5_7 L a ( 5 5 5 £ Y N 1 7 J 5 5
-21 / & ! L \¢ J
¢ 3 3 3 3 ¤ Y & \ ¡ ! ¡ ¢ ¢ 3 3 3
7 m D 1 N J 7! V7 L H £ / F $ CG £ / ¤ 11. 1
L L
m ( D 1 N J 7! #7 L H m ! 1N & & 7
12.
13.
14.
1
3 £ / F
&4 £
/ X £
'65_7 D 1 N J m ( ! m L H 1
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C
C
¢ & 3 £
1L' 3 £ ]J ' % !# £ L / ¢ 4 3 J
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7
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/ ¤
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J 1
5 1L ' ( ]! " 7 -21 . £ J G J $L 1 L)" I+ 1A- H / <3 $ ( 5 & L F J
J
$ 7 ( 7! " ' G $9% J $ L J L] F 3 - 5 1L 7 ¤ 2 1 £ Y H a / <3h
J 1 Q( 5 & L F G J
7 7 3 £ £ / £ / !
mm 5 1
&4 £ £ £ £ £ £ ¤ / 3 C / <3 / m ! ( 5 m
J ] m 5 L H + &- ')a (1 . 5_7 ')( 5_7 G F G9H D £ ¢ <3 6 3 F #$ a ( - 1 ! Y N 5
1 E- a 1 '65_7 L P
15.
+ (
J
5 E- a 1 L
E- a 1 L
3 N1 ,
3 3 3 3 $ 5 1;- E - ( #$ ¤ !
$ 5_7 $ 5 5 1;- E -
1 ¡ < 3 3 J "" ' $ 5 1;- E- L & -21 J $&% ! ')( $ ! $ 5 5 1;- E- L 1 N J
m! 5 m 5 L H 18. D N 1 J m a ')1 . ( 5_7 + &
7 ')( 5_7 D £ ¢ <3 Q G GIH 3 F F # $ 5 E- a 1 L E- a 1 ! ¢ Y N J 1 5 E- a 1 '65_7 3 -; - a 1 L J
Y
1 & 1 3 " " ' $ 5 1;- E- L & -21 ¢ J 3 43 $9% ! ')( $ ! $ 5 5
1- E- L J $ 5 1;- E- #$ ¤ Y N $ 5_7 $ 5 5 ! 1
1;- E-
! £ 19. N 0f f & + 1A- .
7 D F 5_ £ G 3 Q3 CH N 7 1- a a 1 & L
J L J
5_7 Y I+ a 1 . F 3 G N a a
& &
5 3 ¤ F G & ¡\)¡\ 1 3 mm! 7LH 20. D N J 7 m 5_ 7 3 ¢ £ 1 ; C & 3 £ 0& 3 43& = @CB mm ( 21. D N J 7 m ! 5_7 L H 7 ¢ 3 #- 3 £ V1 - ; C & £ 43& #- & 0 = @CB
P N
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m m 5 1 ! #$
N 5 1
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7 7 #7 ( 7 #7 ( 7 ! N 5_7 L _ 5 7 N ! 5 & J
mm 5 ! 5 23. N 5 a
m 1 m .
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a - 5_7 '65_7 ' % a - 5_7 L " ' 3 ¢ J
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$ 5 5 5 1;- ;- #$ $ 1 I ! Y N $ 5 7 $ 5_7<5 - -
m m 5 ( ! 24.
N m 5 a 1 m 5 .
+ 5_ &
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a 5_ 7'65_7 ! -
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& E- a 1 L E- a 1 5 L 3 3 J
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mm! ( D N J m 5_7 m 5 L H a . - 1 ') ( 5_7 D ')( 5_7 3 6+ & G £ ¢ H 3 F a a ( #$ E 1 E 1 ! Y N
a '65_7
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7 $ 5 $ 5_
&- -
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a '65_7 !
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27.
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a 5_7 L $ (* ')( 5_7X! #$ 1;- - $ ( a 5_7 ! ¤ $ 5 7 $ 5 &- E -
mm 5 1! N 5 7 _ 5 7 5
m m m
& a . 7 + &1 ' ( ' ( 5 D ')( 5_7 3 $ G 3 F ¢ G £ ¢ H F #$ E- a 1 E- 5 7 ! Y N
& E- a & E- a & '65_7
a
7 & L 3 43 ¢ 3 ¢ J 5_7 -;
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&<- E- L J
$ 5 $ ( a 5_7 #$ ; 1 ¤ ! Y N
& $ 5_7 $ 5 &- E- $ 5 &<- E-
mm 5 1! ')( 5_7 3 29. N 5 7 _ 5 7 5
m m m
& . ' ( 5 7 + &- a ')1 ( ¢ G 3 F G F #$ E- a 1 E- 5_7 ! £ ¢
N &
E- a & E- a & '65_7
a 7 & L 3 3 ¢ 3 ¢ J 5_7 -I
EL 3 J
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$ 5 ( a 5_7 #$ $ ; 1 ¤ ! Y N
& $ 5_7 $ 5 &- E- $ 5 &<- E-
28.
P 3
+ (
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#$ mm 5 1! 30.
N & m 5 m 5 5 1 ] m 5 . 3 £ £ 3 ¢ <3 £ + & - 3 a 1 ¢ £ ¢ 0 #$ 5 '65_7X! E- a 1 E- 5_7 ! ¢ -; ')( ( n Y N 5 7! _ 4 3
5 5 _ 5 7 ] 6 ' _ 5 7 a
& E- 1 E-
£ Y 3 -
3 " "" ' $ (*')( 5_7X! & -21 ( 7X! $ (*'65n ( 5_7X! ! ) ' ( ! 9 $ % $
#$ $ 5 1 - I- $ ( a 5_7 ¤ ! Y N $ 5 7$ 5
5 ( 5 _ 5 7 a $
& 1;- E-
# $ mm 5 1 ( N & m 5 m 5 5 ! ] m 5 1 .
+ &
- a 1 ¢ £ ¢ £ £ 3 63 3 #$ K3 £ ¢ 0 ( 5_7X! E- a 1 E- 5_7 ! ( -; '6 5n ')( 5_7!
a 5 5 _ 5 7 g ' 5 7 E- 1 E-
3 £ Y -
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& $ 5 7 $ 5 1;- E- 5 $ ( a 5 5_7
31.
Y
N &
P95
Chapter 2
Limits 2.1. Special Functions 2.1.1. The Bessel functions Jν (z), Yν (z), Iν (z) and Kν (z) 1.
2.
3.
4.
5.
6.
7.
a 1 IF G a 1 F G
& 7 ¤ 0& W 7 9/ ¤ 09/ ' 3 ¢ D F O G a 1 & 7 ¤ '
G H
F
63 -;&-21
& * F O'PG a 1 0& 3 a 1 0& ¤ F O'PG a 1 0& 3 a 1 0& ¤ 7 B
&
& M ! a 1
a 1
(:7 B F G -21 0& "
2.1.2. The Struve functions Hν (z) and Lν (z) 1.
2.
. + a 1
. + a 1
F G F G
W 7 ¤
/ - F G W ¤ 7 L 0 / F G H
2.1.3. The Kelvin functions berν (z), beiν (z), kerν (z) and keiν (z) 1.
F G a 1 2D
& P;
7 & H ¤
+ #
2.
(:7X! F G 2- 1 D
, 5
0&
(87!
2.1.4. The Legendre polynomials Pn (z) 1.
2.
3.
4.
5.
6.
7.
8.
9.
7 1 O j F 5_ 7 G C 7 (8 ]! 1 -; £ & -; j ¢ & 7 ]1 ¤ + 1A-; . £ & -; j 0/ & 7 V- \]W ¤ ! ¤ ¢ j ' G 43 1 F O ]! ¤ ¢ 43 1 j a 1 F ' G j F¢ ' G / £ ¤ ¢ j ) * ' , 0& ¤ ' j F ' 5 G 0& ¤ 'g5n 0& ¤ j , ' g ' : 5 ! , )
2.1.5. The Chebyshev polynomials Tn (z) and Un (z) 1.
2.
3.
4.
5.
6.
7 FQ 3 3 ¢ G & 7 £ -; ¢ & 1 ¤ 43 ¢ F 'PG £ & ¤ 43 ¢ a 1 F 'PG £ & ¤ F ¢ ' G / £ ¤ ' 5 £ ¤ ) ' (8 , &
P :
& H
3-"
¤
=
7A@CB
7A@CB =
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
£ & -; -; ¢ ' ) * ' F ' 5
7 ]\ W ¤ / & - ¤ ,
G
¤
FO63 3 ¢ G & F £ 3 £ £ £ ;- ¢ & 1 ¤ 3 ¢ F ' G £ & ¤ 3 ¢ a 1 F ' G £ & ¤ 7 & ¤ ¢ ' F ' G 7 ' 5 & ! ¤ ' ) ' (8 , \]W £ & -; -; / & - ¤ 7 & ¤ ¢ ' ) * ' , 7 ' ' F ' 5 G ¤
3 ¢ G -21
= 7A@CB
2.1.6. The Hermite polynomials Hn (z) 1.
2.
3.
7 '2 ]! ' & F3 ' G F ' ' ( '% &! '!
a 1 F
\]W ¤ - G £ 1 £ & ¤ £ 1 £ & ¤ ' G
2.1.7. The Laguerre polynomials Lλ n (z) 1.
2.
c
- c c F ' G - c £ / c " ' ' % F G d -; c d 3 / dK P<
= @ B = @ B
+ #
( "' ( 8 a ' % F G -; c F ( G
3.
, L
= = @ !C@CB
2.1.8. The Gegenbauer polynomials Cnλ (z) 1.
c
0& ¤ '
d 2 - 1 c 0 &
( O ]! ' ¤ ' % 2. d -; c 0& c 5_7 C7 (8 ! " / (&! 3. = 7A@CB 1A- c& c F G 7 4. d -; c F G ' % 0& ¤ c £ £ & - c F 7 G ¤ 5. 3 & -; c -; c -; & - c 7 ¤ £ ¢ £ 6. 43 & -; c -; c & & c F ]1 - c G ! ¢ ( &! ¤ 7. 43 1A- c c F ' G
O ]! ¤ ¢ ' ( &! 8. 43 1A- c c a F G
1 ¢ ¢ a F ' G 9. 43 1 d -; c d 3 c -; c -;
7 F Q(:7 G -21 £ / Kd & = = #7R@ @CB ¢ ¢ a F ' G 10. 3 1 d -; c 0d 3 c -; c a -;
1 £ d & = = #7R@ @CB 7 ( Q(:7X! ]" / K ( W #$ ' 7 C7 (&! 1 N 1 ! ¤ 11. F G c c -; F ' G 1
1A- c c F ' G ( &! £ & 1 - c & ¤ 12. ' 5
c 2 1
2.1.9. The Jacobi polynomials Pn(ρ, σ) (z) 1.
2.
. -; j + lV- ;- I& ]F 3 . j + l a F ¢ G F 3
,G - V-;&-21 F G ¤ ¤ G P L
5n . g j + lC- a F (* G . -; j + l F G - j + l -; . F ¢ ' G - j + lC-;&- -21 . F
3.
4.
5.
6.
-21 j + 1 lC-;&-21 . ¢ ) -21 j + 1 lC-;&-21 . )
7.
8.
- j + lC-;&- 9. - j + lC-;&10. - j + l 11.
¤ "' & ' % & ¤ g5 7 ¤ 7 5_7! N 5
5 1 1 7!
¤ ¢ ' G
G G F A F a 1L J ¤ F G ' , ¢ ' W F G ¤ ,
4 . ¢ J L G ¤ F ' G ! A F S - -21 . F ¢ ' G 5 % 5_7X! S F G ¤ . ¢ -21 F ' G F 5 G £ ¢ ¤
2.1.10. Hypergeometric functions 1.
- 1 N 1
2.
-R N
m! 5 ^ 1 5
(
` m 5 1 (
!
W
- & W
P P
= @ B - &
= = @ O!C@CB
Chapter 3
Indefinite Integrals 3.1. Elementary Functions 3.1.1. The logarithmic function 1.
2.
3.
4.
5.
6.
Z
Z
Z
Z
Z
' m 5 !
7 m
3 '65_7X! m F]3 G a 1 f O ' % ( 7X! " 3 m ')( $ ! % &- f O a
m 5 !
m Z ' & ' m 5 !A 5 #! '
5 #! Z m 5 ! 7 m 5 ' 5 #! 3 ' 5 #! ' ' ' Z '#" m 5 !
' m 5 ! ( &
m 5 ! m 5 !( & ' 7 ' m ( 5 7 Z m 5( ! m 5 ! 3 m m 5 ! m 5 #! Z m 5 !
m 5 ! (' 5 #!
('%& '
m Z (' 5 #! m 5 !
' 7 Z m 5 ! 5 #!
(' & '
3 ' ' f O 5 #! Z m 5 !
m 5 ! '& ' (' 5 #!
7 m Z 5 #! ' (' m 5 ! 3 ' (' 0f O m 5 ! Z m 5 !
% ' & '
m 5 ! m 5 !' &
Z m 5 ! 7 ' ' 5 ! ' m
3 m m 5 ! ' f m
Z
m 5
N1
m F ¢ G ¤ !
= '8>?7A@ B
!
= '8>?7A@ B
9 = '*>?7R@CB
I = '*>?7R@CB
O
= '*>?7R@CB
Z
7.
m 5 0f O m 0 f O 7 Z 3
& -21 0f O 3 m
31 , Z
m 5
f O ¤
3.2. Special Functions 3.2.1. The Bessel functions Jν (x), Yν (x), Iν (x) and Kν (x)
4-4# $ Z 3 a a 1
1. 2.
3.
4.
5.
6.
= " £ a a 1 ( 9$ 7% ! F G a a ¤ & - 1 Z £ (*'65_7 G &- a a 1
F ( '65_7 Z £ * G F a ¤ '& Z ¢ a 1 5_7X! £ X a 1 ¢ ¢ 3 ¢ X a 1 1 £ 3 1 a 1 ¢! ¤
3 ) ¢ ! ¢ ! ¡ \ ) ¢ & Z 3 3 £
£ ¤ 3 1 Z 3 £
£ ¤ 3 1 7 Z " ! " ! ! F !6G 3 N F !6G = ! !C@MB
7.
Z
8.
Z
9.
Z
10.
Z
! ! F ! G
']" = !C@ !C@ ' & = " '#" = !C@ !C@ ' & 3 =
7
&-21
! F V! G ! ! D " !H ¤ ! ¤ !H ' 7 ¤
2- 1
N ' O' D
N,
= ! !C@MB
11. 12. 13. 14. 15. 16. 17.
18. 19. 20.
21.
22.
23.
24.
7 -; &-21 a 1 3 ' -; ¤ Z " ! ¤ !
Z '& ! 3 ¤ !
' !& ! Z !& !C@ ']" = O' D " ! H ¤ 2 -21 0 = " !C@ '%&
Z 7 ! ! ! 3 ¤ D !H Z 7 !C@ 7 '& ! ! ¤ = " ! !& !
" '#" Z = !C@ ! '%& ' ! !7 ( ! ! ) 7 ! ! Z 3 8 3 8 3 N ! , ! ) ! ,
) 7 Z 3 8 ! 3 8 ! ! ! ,
N ! , ) ) 7 Z 3 8 ! 3 8 !
N ) 3 8 ! , = 3 8 !C@ ) 3 8 ! , Z & -21 (65_7 G - a a 1 £ F Z (65_7 £ G F " ' (
! Z 1a - L' J
(65_7 Y Z a 3 £ &- F 3 G -; &1 Z V <3 # 0
#
1 3 D # <3 #1 3 7 Z " ! " ! ! D ! H 3 N D ! H Z
N 3
! ¤ !, = ! !C@MB = ! !C@MB = ! !C@MB
-; a ¤ a a ¤ - 1 7
1 H ¤ = ! !C@MB
25.
Z
26.
Z
27.
Z
28.
Z
29.
Z
30.
Z
31.
Z
32.
Z
33.
Z
34.
Z
35.
36.
37.
38.
31 ,
7 &-21 2- 1 ' ¤ 7 -; &-21 a 1 ' ;- ¤ '& ! ! <3 ¤
" ! !
7 ! 3 N D ! D !H
'#" = !C@ !C@ '%& ' ! = = " ! 3 3 !
!& ! 3 !
: &-21 2- 1 3 Z -; : &-21 a 1 ( 7X! '%& Z a 1
Z
- I B ;
9 B ;
¤
3 £ X 1 0 ! !H
¤
= ! !C@MB
7 ! !;( ! !C@IF !#G ¤ ¤
¤
7 ' : ¤ 7 ' 3 -; : ¤
43 ¢ a 1 3 ¢ ¤ 1 I B ! 7 5 5 Q(:7 9 3 ¢! :3 ¢) I 3 ¢! ) 3 ¢ 3 6 65 5 Q(:7 1A- 0 = 7 g5 ( P5 7 I 3 ¢) ¢) ; 9 3 ¢) - ¢ 65 ( P5_7 1A- 0 =
39.
9 ¢) 3 ( ¤ N95
@CB @CB
3 E5 7 3 ( 2- 1 S 2- 1 7 9 ¢) 3 3 ¢ S £ 4 3 ¢ 2a S 1
40.
3 2- 1 0 S = >?7A@CB
3.2.2. The Struve functions Hν (z) and Lν (z) 1.
2.
Z
& -21 7( £ F 7 & -21 3
( G "" '& 1;J g5 (* $
- 2- 1 & " " - L" ! ( $ 5 1 L 7 J ( ( Z £ G F ;- L -; ¤ " ( Z
L & -21 2 &- a 1 L &- a 1 J ')( $ ! % L - -21 " ' % ' & & -21 ¢
L 3 " 3 ')( $ ! % O'T(* J $ 5_7! ( $ 5 1L J ' ] " !& £ ¤ a 43 &- 1 L -;&-21 <3 * ( '65 1 L J
Z ¤
H
H1 H 3 Z L 3 L ¤
L
1 L
-
L
3. 4.
3.2.3. The Airy functions Ai (z) and Bi (z)
4-4# $ f f - %03 $ ,- $ '= Z 1. &-21 3 3 ¢ &- 3 ¢ R 3 £ 2. 3. 4.
&-!&
= = @ !C@CB
= = @ O!C@CB
Z
Z 2 3 Z 2 3 £ &
Z
= = @ !C@CB £
Z
= = @ !C@CB
N;
5.
Z
6.
Z
7.
Z
8.
Z
9. 10.
&
31 , 3
= = @ O!C@CB
3 2
7 '65_7 D a 1 3 & -21 ' ¢ £ 3 R 3
3
Z
= = @& C7X O!C@MB
7
2 3 7 Z 2 & 32 £ Z
' 3 3 ¢ &-
&-!& H = = @ C7O7X!C@CB
= = @ C7 !0!AB
3
= = @& C7 !C@MB
& &-21 7 ' (87 Z 3 & - & 3 &- & 3 ¢ R 3 £ & -!& &
£ Z @ !C@CB 3 & -!& &
==
Z
& &-21 7 3 3 ¢ ¢ &- & 3 7 3 3 £ R 3 &- 7 £ Z 7 £
3 &-!& & 3 3
11.
12.
Z
&
13.
Z
& &
14.
Z
&
&- & 3 £ &-!&
7 £ 3 R 3 R 3 V &- & Z X 3 R 3 V R 3 # &- &
= = @ !C@CB 7
3 = = @& O!C@MB & 3 & 7 7 Z
3 & 3
= = @& !C@MB & &
3
7
15.
Z
& 3
&
& 3
& & 3
&
3
Z
= = @ O !C@CB
&
& &
£ 2
N :
3
& 3
&
= = @ O !C@CB
16.
Z
Z
18.
Z
19.
Z
20.
Z
21.
Z
22.
Z
23.
Z
24.
Z
25.
Z
26.
Z
27.
Z
28.
Z
7 a O ' 5 1 3
£ & -21 & & -21 3 ' 3 ¢ &- ' ¢ £ Z Z 3 R 3 &-!& 3 3 ¢ &-
= = @ !C@CB 7 a ' 1 3
3 &-21 & ¢ &- & 3 7 3 ¢ R 3 & -
&-21 3 3 Z 3 ¢ R 3 £ &- 3 3 ¢ X 3 £ R 3 V &-
Z 7 ¢ £ 3 R 3 R 3 &-!&
= = @ !C@MB 7 £ @ &
2 3
= = !C@MB & 7
7 F & 3 & 3 & 3 G @ = = 7X!C@CB Z
3 &-21
= = @& C7 !C@MB
17.
7
7
7 7
)F 7 ' & 3 ' 7 & 3
= = @& C7 C! @MB = = @& C7 C! @MB
7 O '65 3 ' 3 £ &-21 7
3 2 7
F3 &
Z
&-21 &
= = @& C7 C! @MB = = @& C! @MB
Z
&-21
= = @& C7 !C@MB
3
a 7
3
£
2
N<
G
a 1 ¢
£ £ Z & -
3 G
= = @ O !C@CB = = @& 7!C@MB = = @& O!C@MB
29. 30.
31. 32. 33.
34.
7 2 3 & 2 3 7 ' 7 Z Z 3 & -21 & 3 ' 3 7 Z 7 & & 3 &
& & 7 Z
& 7 ' 7 Z Z & 3 7X &-21 3 ' X 7 3 3 ' Z & 3 a 1 & 3 &-21 ' ¢ 7 Z # & 3 3 R 3 Z
35.
Z
36.
Z
37.
Z
38.
Z
39.
Z
40.
Z
41.
31 , 3
Z
3
&
&
3 2 &
&
3 & & 7 &
'65 Z 7X 7 '65_7X! 7 a 1 3
7 &
7
3
£
3
a 1 &
¢ Z & - &
= = @ !C@CB = = @& C! @MB = = @& C! @MB
a 1
¢
Z
= = @ C! @CB
&-
'
3 ¢ &- & £ Z &-!& &
= = @ O!C@CB = = @& 7!C@MB
7 7
= = @& !C@MB
Z
= = @& O!C@MB
&
& 3
7 &-21 & 3 ' ¢
3 3 R 3
= = @& !C@MB
£ &
' a 1 3 ¢ &- £ Z &!- &
= = @ C! @CB a & 3 a £ £ a 1 & ¢ £ ¢ Z R 3 a
' ¢ £ Z @ R &-21 = = !CC@ B
3
'
2 3 7
! ! F !#G N F
&
7
3
&
Z
! !#G N L
&
7
7
7
&
= = @ O !C@CB
= ! !C@CB
Z
42.
7
! EF
! !VG 3 N F
! ! G
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3.2.4. Various functions 1. 2.
(' ( !" 3 ' % (:7 3 ¢ ') ( $ ! % - K3 3 ¢! ¤ Z 3 £ 0 2 ¤ 2
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7 Z j j 3 )( !A 65 5_7X! Y j j <3 j j -21 <3 83 4 j j 0 ¤ 2 1 7 Z j 0 3 5_7 " ! ! ¤ Y Dj
j j j j H h 3 3 -21 -21 Z
3.
4. 5. 6.
7.
8.
Z
9.
Z
2 3 3 2 7 F 2 n3 :3 E
7 F
K
N P
¢
K
2
3 £ E ¤
G
%
K
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7
3
G
E
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Chapter 4
Definite Integrals 4.1. Elementary Functions 4.1.1. Algebraic functions
; $ # $ f
1.
¡¤
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1
2.
Z
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( ( / J ( ! 1 L F 3 G a 1 = m
3.
4.
5.
Z
Z
Z
( 7 !
!
( m ! @ B
( m ! @CB
3 0fg3
-21 0fg3 -21 ¢
¢ 3 m j ( m , ) ( m , )
3 0fg3 1 f -21 0fg3 1 ¢ 3 0fg3 1 £ -21 0fg3 -21 ¢
E
m , )
m , E )
=m
@ B
& ( m ! @CB =
& ( m ! @CB =
+
6.
7.
8.
m m 3 0fg3 -21 K ) , 3 E ) 1 0fg3 1 ¢ & ( = Z m 3 -21 ¢ 3 0fg 3 -21 £ K ) , -21 0fg & ( = Z
Z
10.
11.
12.
13.
14.
, m ! @CB m ! @CB
3 0fg3 -21
-21 0fg3 -!&4 ¢
9.
5)
5 m
K
m 5 m
=
Z
3 0fg3 -21
-21 0fg3 -!&4 ¢ ¢ m 1 ¢ 3 m 1 ( m ) , , ) -21 fT3 -21 ¢ 3 0f63 -21 ( m Z ¢ ¢ 3 1 3 0f63 -21
3 1 0fg ) Z
Z
3 0f63 -!&4
1 0fg3 1 ¢
( m ! @ B
( m ! @CB =
=
( m ! @CB
m -21
,
3 ¢ & ( m ! @CB =
m m ( m ! @ B E , 3 K) , = ) " m ( ! " Z £ K m 3 £ D m <7 5 7 ( m ( !
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( m !
!,
15.
16.
m ( ! m ( m ! 3 E , = 7<5 7 ( m ( !
) " m ( ! Z m m f K) 7<5 7 ( m ( !
, 3hf D ) , & ( m ! = 5n ! Z 0fg 3 O £ - -21 / f 4a &4 5 3 4a 1 ¢ 0f6 L J
( ] 5n ( 7 5 m !
Y N 5 !
1 ! ( =
Z -!&4 fT3 -21 ¢ 0fg3
5 / £ F ¢ m G j m 5 m , = & 5 m ! ) Z
@CB
@CB
17.
18.
19.
20.
21.
22.
Z
3 fg3 1 ¢ 7 m ¢ m F
@ B
@ B
1
m ¢ m G F 3 G
5 m 6( m
m 3 ¢ = 5 m ! @CB
5 m 6( m
= & ( m ! @ B
Z
3 0fg3 1
-21 ¢ 7 m / G f F¢ 3 *
Z
3 fg3 1
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Z
3 0fg3 1 £ &4
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3
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m , ) * = ( m ! @CB
+
23.
24.
25.
26.
Z
3 fg3 -21
1 ¢ 7 ¢ m F G
5 m m
6( m 3
-21 ¢ 0fg3 -21 *
¢
2 1
3 0 g f 3
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Z
Z
= & 5 m ! @ B
m *
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5 m 6( m
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27.
Z
28.
Z
3 0fg3 -21 £ &4
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3 fg3 2- 1
1 ¢ m m ( m !
29.
Z
30.
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31.
5)
3 0fg3 2- 1 -21 ¢
K
m ¢ * 3
( m !
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¢ 3 F ¢ m * G -21
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5
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m 0fg3 -!4& m 5n
32.
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-21 ¢
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3 fg3 -!&4
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£ 3 * ( m ! 0f 3
K
m , £ ) *
34.
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¢ 3 !- &4 fT3 -21
35.
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3 fg3 -
1 ¢
36.
37.
38.
m , ) *
0fg3 -!&4
( m ! m (:7X! ( m ! m
Z
E
3 0fg3 -
-21 ¢
( m ( m ! m
= 5 m ! @CB
= 5 m ! @CB
( m
m , E ) * = ( m ! @CB
m *
= & ( m ! @ B
m *
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m 3 fg3 - ( m ! -21 fT3 1 ¢
Z
= ( m ! C@ B ( m! Z ¢
2 1 T f 3 3 0 g f 3
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m ( m ! ( m !
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39.
Z
40.
5 1L C7( ! "
J _ 5 7X! N 1 5_7 m L C7<5n m 5 m ! C<7 5n 5 !
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3 -21 ¢
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1
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+
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; $ # $ f
¡¤
W
£ -; - + a 1 . W + 0.
F G
1.
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2.
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1
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3.
m
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4.
Z
5.
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6.
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7.
Z
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8.
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1 0fg3 1
9.
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10.
2
+
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Z
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m ¤ ) ,
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f) 3 7 , &4
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fI
fI
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5: n 5 ! 5 L 1N 1 5 J
m . 1 / + - f 3 ¢ *
13.
Z
14.
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m . -21 fT3 1 / + -
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£ - 2- 1 / f9 4a &4
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18.
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£
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@ @CB
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; $ # $ f
1.
Z
¡¤
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& & 9a a 1 9a 1 5 7
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3.
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m m 0fg3 1 F G ¤
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0f63
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1 0fg3 -21
7.
Z
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8.
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9.
Z
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4.
-21
Z
-21 fT3 -21
11.
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m m 0fg3 L F G ¤
m 0f63 L F G ¤
10.
fQ3 m
m G D £ L1 F * G9H ¤ m m G 3 £ L1 F GIH ¤
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m ¤ G V
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1
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17.
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18.
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m
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m F1 #" G ¤
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m m m fg3 D / £ F #" G / f & F #" G9H ¤
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19.
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20.
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L
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m ¤ G F 2- 1 *
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22.
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2
1
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24.
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1 0fg3 1
25.
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1 0fg3 -21
26.
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27.
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28.
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29.
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m - ) * , ¤
m m m 0f63 D £ 1 F G f F GIH ¤
m m fg3 F G ¤
m fg3 F G ¤
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m £ m m ¤ G G 9G H F D F f F * 1 1
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m m ¤ F G
m ¤ G V
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30.
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31.
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m D f L F " m G 3 / £ f L F " m G -21
32.
Z
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33.
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1 0fg3 -21
34.
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35.
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36.
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L
L
m F1 " GIH ¤
m ¤ G F # " 2- 1
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£ 3
7 D + . F 7 G 3 + . F 7 3 G9H £ <3
7 D + a 1 . F 7 G
. 7 + a 1 F 3 G9H
m
m ¤ F #" G
G ¤
=
7 X@CB
=
7 X@ B
4.1.4. Trigonometric functions 1.
Z
2.
Z
m ! m fI L fI 0fI L fI 0 5 7
2- 1 1
'
5
'
'
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=m
@CB
+
3.
Z
4.
Z
5.
Z
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7 (:7
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7 (87
6.
Z
7.
Z
8.
Z
9.
Z
fI -21
O'65_7X! @ =
= '6 5_7X! @
Z
Z
12.
13.
Z
-21 fT3 -21
1
;
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!
7A@ B
!
7A@ B
=m
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@ B
'
¢ ; ¢ ¤
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£ &4
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f
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£ 2- 1
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f
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5) 5
f /
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m m 1F G
fg3
3 Ff
fg3 / = = @ !C@CB 7 7 '
!
FOf m m G * m m m 3 G gF G 3 * !, ,
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@ B
m F *G m
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@ B
14.
15.
16.
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Z
Z
-21
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£ " D m F m * G 3
0fg3
m F m G H F m G 1
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1 0fg3 -21
18.
Z
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19.
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20.
Z
Z
m F1 GIH
=m
21.
- fT3 -
22.
Z
0fg3 4a 1
fg3
@ B
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@CB
@CB
@ B
H
F G
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0fg3
H
m F G
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m G
F * 1
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0fg3
@ B @ B
H
m GIH F1 *
=m
@CB
0fg3
m m G G9H = m @ B D
F
F * -2 1 &4 £ - -!&4 / f 4 a fg3
W #$ 5 5 L ( ( @ B
! m J 5 ! N =
! 1
& ] 5
Y
H
m m £ D f H F G 3 H1 F G9H m m
Z
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m
F m G D£ 3
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£ F m * G H F m * G F m * G D 3 1
m m TF *9G H
17.
!,93
+
m m m fg3 D / £ F #" G 3 / f & F #" IG H = m @CB
23.
Z
1
24.
Z
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25.
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-21 fT3 1
m fg3 f 1 1 F#" G
Z
-21 fT3 -!&4
27.
Z
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28.
26.
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Z
-21 fT3 -21
29.
30.
Z
31.
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2
m fg3
fg3
H
m F1 #" G
£ F#" m G m m G H1 F #" G9H
@ B
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m F #" G
H
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@ B
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f 4 a 2- 1 Z
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m H F #" m G 3 m
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5) 5
W W
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m H F G -21
m
H
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m G F 2- 1 *
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m £ m m £ G
D f H F 3 f ] H F 1 G f H !,!5
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32.
Z
-21
fg3
£ " D m F m G
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1 0fg3 1
34.
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1 0fg3 -21
35.
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-21 fT3 -21
36.
Z
-21 fT3 -21
37.
Z
38.
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39.
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0fg3 4a 1
m fg3 F *G
m m G D £ 1 F G h 3 f F IG H / £ m F m G fg3
@ B
=m
fg3
1 F
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@ B
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Z
m m TF G9H
m m m G 3hf F * G9H fg3 D £ 1 F * = m @CB m m fg 3 F G = m @ B
33.
m
fg3
m G
F * -2 1
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@ B
@CB
=m
@ B
fg3 £ - 2- 1 / f9 4a &4
5: ( W #$ 5n ! ! Y = m ! ( 7A@ B 1 ] 5 5 L 1N
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Z m 1 fg3 Y D f H F #" m G 3 / £ f H F #" m G H F #" m IG H = m @CB -21 1
40.
41.
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fg3 / f !, ;
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m G F " 2- 1
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@ B
+
42.
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1 0fg3 -21
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-21 fT3 -!&4
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44.
Z
fg3
+ m F #" m G 3 m F #" m G
1 m fg 3 / £ F #" G
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@ B
@ B
a 1 5_7V( ( #$
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7 a ! _ 5 7
( 7 C7,5 ! @ B = !
£ R ¢ f
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1
=
C7<5 m ! @ B
=
C7<5 m ! @CB
X ¢ f
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5_7 L ( ! J
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f " 65_7X! 0f a 5_7 L f - 5_7 L J
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a ( 1 1 '6 5_7 ! ( m
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5_7 ( ( #$
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49.
50.
51.
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f 0f
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7
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3h
m ! m
f
0f m ! m
54.
Z
Z
f £ £ N f &
#$
¤
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1
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m G ES-21 a 1
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@ B
7 5 (* $ 5 ' ( 7 #$ = m @CB
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+
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
5) 5
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V-
m 7 ( # , 7 5
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m
&- & W #$
( " SZ 7 ( 7( N ! #7< 5 ¤ -
m 7 (
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N 7 ( #7,5
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0f 1 V7 m f £ £ N ¤ ! 7 ( V < 7 5
m
f
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m 1 &- &
W #$ 7 ( 7 Z = m @ B #-
N 7( ! #7, 5 m 1
W #$ " V7 ( Z 1 !
= m @ B N & #7 ( #7, 5 m
&
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-
66.
Z
67.
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68.
69.
70.
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( 1;- ! m
f
0f m ! f £ £ N m f 1 7 (
W a 7 N m 7 (
= m
@ B
Z
SZ
SZ
SZ
SZ
7
7
f
0f m ! m
f £ £ N f &
f / 0f m ! f f m
f 0f
/
m ! m
!,9P
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f f
G =
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W #$ 7 ¤ ! V7< 5
W #$ 1 V7 ¤ ! & #7 ( V7< 5
£ £ N
5_7 #$ ¤ ! a 5_7
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74.
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81.
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82.
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85.
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86.
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87.
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88.
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89.
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90.
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91.
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#7 #7 ( #$ 1 ! 5_7 N
m & &- & a
m & C7<5 ! ! $ # 1 1 #7 #7 ! ( N m & & V7 ( #7<5
& C7<5 ! m ! X" J L F m G 3 ] M 7 (:7X! 3 m
= m
@ B ¤
B B
@CB
m & 9a 1 L
5_7X! J L J
9a 1 #$ ( 7 C7<5 m ! @CB
=m ( m ! ! !
1 #7 9 a 1 N & 5_7 9 a & &
7,5 7( Z1 145. -21 7% ( 7( 0f
#$ 9a 1 #7 9a 1 m J 1 L ¢ N 3
5 7! L & & 5_7 ! ( m J
7<5 7( Z1 f
7 ( 7( 146. m f9 IF 7 7 £ ¢ 7 ¢ 7 f * f 3 * f )
Y
0f £ £ f
5) :
7 9a 1 9a 1 #$ & N 9 a & ! ( m = m ! ( R7 @CB
!< ,
,
f ¢ 3 £G = & C7<5 m ! @CB
Z
147.
148.
149.
150.
151.
_ 5 7X! (87
Z
-21 f9 3 2
-21 7 f 4a
Z
Z
Z
153.
Z
m
3 m ¢ 3 3 m F
f9 32
-21
-21 fg3 -21
F ¢) G N & m
7 m D 2 F GIH
m G
= m
@CB
7 #7#7 5_7 m #$ ! & 5 I5_7
( C7( m ! B ! !
¢ 3hf9
3 ¢ H = m ! C 7( m ! @ B
0 f9
m
f9 32
1
m D f9
<3 F 7 3 ¢ G m
f 4a a 1
152.
m (
m
=m
!
C7(
m ! @ B
0fg3
¢) ¢ N m
!
W W 7#7 #7 _ 5 7 I5_7 #$ ! & 9a _ 5 7 Ia a &
( 7 ( m ! B !
1 fg3 1 fg3
7 f9 3hf9
¢ 3 m 9f m ,
) , ) ( m ! @ B =m ! Z m m 3 -21 fg 3 1 fg
) , = m ! ( m ! @ B
!<93
+
154.
155.
156.
157.
158.
Z
5) :
1 fg3 -!&4 fg3
£ f 5( m ¢ 3 m 3 , m
) m= Z 3 -21 0fg 3 -21 fg =m
Z
-21 fg3 -!&4 0fg3
f 5 ( m £ ¢ 3 m , m m )
m
) !
!
m
)
,
( m ! @ B ,
( m ! @ B
Z
-!&4 fg3 -!&4
f 5( m
0fg3
m £ m ¢ , m ) 3
=m
=m
!
& ( m ! @CB
!
& ( m ! @CB
fg3 4a 1 fg3
W #$ 7 #7V7 5 " " m & 5 ! ! N &
5 L & ] 5 J
= m ! ( ! ( m ! @CB
Z
159.
160.
Z
1 fg3 -21 fg3
m m ¢ f £ h 3 f
) 3 , f ) , = m ! & ( Z m 3 !- &4 fg 3
-21 fg ) = m ! & (
!
,
m ! @ B m ! @ B
161.
162.
163.
5 m
fg3 m 6( m m ¢ & ( m ! m ) 3 , =m ! 7 Z 5 m ! 5 m ! 5 m
- 7 7<5 7 ¢ 3 £
= m C7( !
7 ( ! m Z -21 m 5 m (
7 #7V7 5_7 5_7 m #$ 5_7 L ! ] 5n O! f 4a J N n 5 L J = m ! ( ! & C7 ( m ! S Z " ¢ S m 3 V- #$ 1 #7V7# 7 #7 ! & C7 ( ! Y N = m
# 7 ( # , 7 5 ! &
" V7#7 #7V7 #$ Z 1 ! m N & V7 ( #7<5
m & C7 ( ! ! Z m 5 m ( m f £ ¢ 3hf <3
7,5 m C7 ( m ! 7( m = m ! Z
- fg3 -!&4
@CB
@CB
164.
165.
166.
1
167.
Z
168.
Z
7,5 7 (
m 5 m (
3 7 D f ] f9 3
@CB
@ B
B @CB
£ 3 ¢ ¤
m f
7<5 m ¢ 3hf9
<3 £ 7 ( m ]f 3 ¢ ¢ f & & H = m ! C7( m ! @ B !< ;
+
1
169.
Z
170.
Z
7,5 7(
5) :
V ¤
£ 3
5 m ( m
D 0 f O 3 4 3f O <3 7,7 5( m H 3 ¢ 3hf9
m m
m= ! C7( m ! " " 5 ! Z
f
5: L a 5n L - J
J
7 #7V7 a 5n #$ & ! Y N = ! C7 ( ! & - 5n a 5n
" " 5 ! Z 0f
f - 5: L a 5n L J J 7 #7V7 a 5n #$
£ & 0 f ! Y £ N
n 5 a f - 5n &
= ( 7 ! C7( ! SZ 7 f
f #$ 1 V7#7 #7#7 ! m f £ £ N m f & ] #7( V7<5
C 7( ! = 7
@ B
171.
@ B
172.
173.
1 7
174.
Z
175.
Z
7<5 7 (
@ B
@CB
3 £¤
1 2- 1 0f
7#7 #7 9a £ Y ' N & & ] 5n !
m 9 a & L 5: O! J 5n L #$ J 7 #7 5_7 9a & & £ m 3 N & & 5n 5: ! m ( 7 C7(
!<9:
!
#$
m! B
1 7
176.
Z
177.
Z
178.
Z
1 7
1
f & f
¤
0f 7 m f9 0f9 3 ¢ 0f9 0
1
Z
179.
&
C7 ( m ! B
C7 ( m ! B
f
7X Vf9 ¢ # f 3 & f9 3 ¢ 4 ¢ h 3 f9
¢ £ f 3 ¢ f9
m C7 ( ! m
1
180.
Z
181.
Z
&
1 -21
Y
182.
1
Z
183.
1
Z
¢ & N
0f
m L 5_7X! J I a & L #$ J 7 9a 5_7 #$ 1 3 & N 9a 9a ( m & & ! ( 7 C7,5 ! m B !
1 V7 5_7
& 9a & ! ( m
0f
m
)
¢ f 3 7,5 7<5 m 3 ¢
, C7<5 ! m
0f
m 3 f 7 m 7,5 m £ f f 3 ¢ 2
B
7
7 ¤
B
!< <
m
7,5 ,7 5 C7<5 m
m !
B
+
Z
G £ 184. F £ ¢ £ ) 3 7X , 3 7
5) ,
£ 3 7 £ ¢ 3 , 3 ) 7 ¤ 7X
&
4.2. The Dilogarithm Li2 (z) 4.2.1. Integrals containing Li2 (z) and algebraic functions 1.
2.
3.
Z
-21 0fg3 -21 0fg3 4
W #$ 7 V 7 # 7 _ 5 7 I5 _ 7
f 4a a 1 ¢) ¢ N ! Ia 5 7 9a a &
( 7 & ( m ! B m ! ! ( ( m Z
f ) 1 0fg3 2- 1 fg3 4
, 5 ( m m & ( m ! C@ B m 3
= ! ( ( m Z £ 3 4
) -21 0fg3 -21 fg , 5 ( m = m & ( m ! @CB 3
!
4.
5.
Z
-21 0fg3 -!&4 fg3 4 ¢ 3 ¢ 3 m 5 m *
( m
=m
!
& ( m ! @CB
" " m & 5 ! 0fg3 4a 1 0fg3 5 L
J
7#7 #7] 5 Y N ] 5 = m ! ! & ( m ! @ B &
!
Z
!<9L
+ #
6.
7.
8.
!
Z
-21 0fg3
Vf 1 m £ 3 ¢ * * m )
Z
m £ * 3 , = m ! ( m ! @ B
-!&4 0fg3 - 0fg3
¢ 3 ¢ 3 m ¢ ¢ 3 m 3 , m * * ) m=
Z
-21 0fg3 -!&4 £ 1 £ 7 3
)
£
!
( m ! @ B
0fg3
7 m * ¢ 3 , 3
7 7 m
) * ¢ 3 , = m ! ( m ! @ B
4.2.2. Integrals containing Li2 (z) and trigonometric functions 1.
2.
SZ
Z
7
f 0f
m m
1 #7 #7#7 #7 £ £ N #7( V7
= m ! C7 ( ! " " 5 ! f
0f a 5n L - 5: L J J 7 #7V7 a 5n #$ £ & 0 f ! Y £ N
n 5 a f - 5:
= ( 7 C7( ! f f
!
3.
@ B
Z
f
" " 5 ! N - 5: L a 5n L J
J
@ B
!<9P
7 #7#7 a & ] - 5:
= (
5n #$
a ! 5n 7 C7( ! @ B !
+
4.
SZ
N m
Z
6.
"
"
Z
5.
1 1 # 7#7 !
& - & a
1
& &
"
¢ 3 m
N m
#$
#$ 1 V 7#7 # 7V7 ! ¤ #7 ( # 7,5
-S
5) , 3
N
m 5_7 N
1 V 7#7 # 7V7 ! #7( # 7,5
7 #7#7 #7 & #$ ! &- & a m ! C7 ( ! B #$
m
!
& C7 ( ! B
4.2.3. Integrals containing Li2 (z) and the logarithmic function 1.
2.
3.
m & !
5_7X! 5 & L
J 7V7 5 7 7 5_7 5_7 m m ¢ Y N ! ! 3 N 5 n 5 5
&
&
& &
7 #7 #7 5_7 m ( C7( ¢ ! m = #' N & 5 & ! !
Z m 5 m ( F £ f 7 G / f I
7 ¢ & C7 ( f * m 3 / f I 3 f = m ! Z m 5 m ( -!&4
/ f / f g3 7<5 7( m £ / ¢ 3 f g3 £ m ) , m= C7( Z
-21
m 5 m (
!
m ! @CB
m ! @ B
m ! @ B
4.2.4. Integrals containing Li2 (z) and inverse trigonometric functions 1.
1 7
Z
/
X7
&
£ 3 £ £ ¢ £ ¤
L N
!>
2.
1 -21
Z
!>
m 5 & L
! 5 7J ! 5n
f
7 5_7 5 Y N 5n 5n & 3 ¢ N
m &
&
! 7 #7 #7 5 ¢ #' N & 5n /
7 #7 5 & n 5 m ! & = C7 ( m ! @ B m!
4.3. The Sine Si (z) and Cosine ci (z) Integrals 4.3.1. Integrals containing Si (z) and algebraic functions 1.
m m 0fg3 £ / f F #" G 3 £ m m m / f D 1 F #" G H F#" G 3 F #" G H1 F #"
Z
-21
/ £ 1 F#" m GIH
m
G
=m
@CB
4.3.2. Integrals containing Si (z) and trigonometric functions 1.
7
Z
£
m ( !
£
m
D fI
2.
3.
m ( !
Z
£ 3
f
4.
" " 5n ! - a & a a & J L J
Z O' ! 0f
m
m m F G m 0fI H F G
m m 3 F G m m m D fI <3 0fI H F G
7
Z
@CB
=m
=m
£
N L &
5 7 1 _
& & - a
W a & (
& ! a a &
#$
=
@CB
( @CB
W #$ ' 5 " '#" m '%& 1 #' 5 1 ! ( ' 5 7! % ' 5 7 ! N & ' 5 # ' 5 # O' 5 7 ¤ & &
L1
+
5.
Z
f 0f
Y
6.
7.
8.
9.
10.
11.
'
Z
!
f
&N
" " 5: O! a a & L - a & L J W
J
1 5_7 a & ( #$ ! = & & - a & a a &
f £ £ 0f
"' ' & '6 5_7X! % m '65_7X!
SZ
Z
7
Z
7
"
0f 0f
m
m
f f
Z
"
£ £ &N
£ 3
£
¢
m 3
m &N
-S
¢ f G
= m
F f
¢ f G
= m
&N
@ B
W #$ 1 1 #7 ! ( ¤
& & #7 ( # 7,5
W 1 1 V7 ! (
& & #7( V 7<5
L,
@ B
C7<5 m ! " 3 " m 5 7<5 m ! -
W a 1 ( #$ ¤ ! a & 'g5 7
a 1 a &
( X@ B
W 1 1 7 ( #$ ¤ ! & & 7, ( 72 5
£ 0 m 5 7<5 m ! 3 " f C 7<5 m ! F
S Z "
' N &
£
V-
5) 31 ,
#$
= m
@CB
!> Z
12.
!>
" " C7 ( !
7( 5 7 ( ( Y F G F G &N
W #$ ;1 - a ! ( 1- # 7 (
=
1 1;- & &
7R@CB
4.3.3. Integrals containing Si (z) and the logarithmic function 1.
Z
-21
Y
2.
Z
Z
¢ N & m 5 m (
m 5 m (
&
3.
& m 9a 1 L 5_7X! J 5_7
L J W W W W #$
#$ 9a 9a 9a 1 1! ( 1 1! ( 3 N & 5 7 9a
& & 5_7 & &
= m ! ( 7R@CB m £ f # f 0f O 3 0f O
1 ¢ 0f9 3 1 f O H 0f O 3 0f O H1 f O 0 = m @ B
m 5 m (
m ] 1 0f O £ f # f ]0f9 ¢ 0 f O 3 f9 m f 2 1 f O H 0f O 3 0 f O H1 f O 0 = m
@CB
4.3.4. Integrals containing Si (z) and inverse trigonometric functions 1.
1 -21
Z
0f
m 5_7 L ] 5_7! J 9 a & L J
3 £ f
m
2.
1
Z
&N
1 9 a 1 5_7 ! (
& & 9a & 9a &
W #$
=
( 7A@ B
£ 0f9 ¢ fI 3 £ f fI 1 ¤ f9 ] 1 fI H 0fI 3 0fI H1 0fI L 3
+
5) 31 ;
4.3.5. Integrals containing products of Si (z) and ci (z) Z
-21 £ 3 £ 0
" " V 3 ¢ 0
7<5 =( D F G 3 F 9G H
1.
2.
Z
3.
Z
-21 £ 3
@ B
£ 00 £ £ 0
£ - -!& 7<5
( 7 R7 @CB D F G 3 <3 F GIH = W 1 1 #$ ! m ! / E / fg3E = m @ B
N & & &
4.4. The Error Functions erf (z), erfi (z) and erfc (z) 4.4.1. Integrals containing erf (z) and algebraic functions 1.
fg3 f 4a F W W #$ 5 I5 ( Y N 1 9a 1 a I 1 a! 5 7 =m ! & & & 1
Z 4 a 1 0fg3 0fg3 £ - -21 f 4a W #$ ] 5 ( Y N 1 5 ! f ¡
&
Z
-21 0fg3 -21
7 7 G
2.
3.
Z
1
5 L J 5 !
0fg3
" m - W f m 0f ¢ m , , 1) )
( 7 X@ B
3
¤
L95
=m
@CB
4.
Z
! #
-21
" f - W m m , , 1) ) = m @CB
fg3
(
4.4.2. Integrals containing erf (z), erfc (z) and the exponential function 1.
-21 0fg3 -21 W + - .
Z
W
4a f .
=m
=m
4.
Z
-21 0fg3 -21
5.
Z
4a 1 0fg3 W / + - .
W
.
+ -
7.
Z
Z
1
-21
W/
W/
+ -
+ -
W
.
.
+ -
Y
6.
0fg3
F m G m
@ B @ B @ B
£ - 2- 1 Of9 4a 0fg3 W #$ 7 5 5 L ( @ B
! m J 5 ! N =
!
& 5
.
( 7 @ B
m 0fg3 F G
-21
Z
!
3.
=
W W #$ 1 ;5 1 ! a I a 5 7 1
m
=m
Z
+ -
0fg3
WW 0fg3 F 3 ¢ G
2.
7 5 7 7 F G & N & 9a &
0fg3
7 " D £ h £ f 3 ¢ W H 3 f
=m
W 0fg3 F 3 ¢ G
=m
L;
@ B
@ B
+
8.
W . -21 0fg3 -21 / + -
Z
5) 5) 3
m 0f63 m F #" G = m @CB
4.4.3. Integrals containing erf (z) and trigonometric functions 1.
Z
2.
Z
0f
'
!
f
7 ( m
#$ 1 #7 ! ( ¤
N & - & a
m !
"' m '%& '65 7! _ 5 7 J L
#$ a a ' 1 1! ( m ¤ N a ] '6_ & 5 7
Z
3.
' !
6 " ' m '%& '65 1 L ' 5 1 '65 1 ( ! ' % O'65_7XJ ! N
' 5 & ] '6 5_7 g
" " 5n ! - a & a a & J V7< 5 L aJ #$ L & ! 1 Y N =
a & & & - & a a &
" 5n O! - a & L a a & L J J #$ #7,5 a £ & ! ( 1 f = £ N
a a a & & 0f & - & &
"
" 5n ! W W
- a & L a a & L J 7#7,5 J a
#$
& ! Y N = & & & - a & a a &
0f
Z
4.
5.
Z
f
f 0f Y
6.
Z
f
m #$ ¤
( @CB
( X@ B
L :
( @CB
7.
Z
! #
f 0f
Y
8.
Z
9.
Z
W a
-
-
W
" 5n O!
a a & L - a & L J
J 7 #7<5 a & #$ N ! = ( 7A@ B & & & - a & a a &
#$ 1 # 7 ! ( m 5_7 = m @ B
N &- & a
7 #7 m 5_7 N &- ! & a
= m @CB
W
f £ £ f
W
(
4.4.4. Integrals containing erf (z) and the logarithmic function 1.
Z
-21
m 5 m (
Y
&N &
1 9a
&
m &
5_7X! J 1 Ia 1 ! ( 5_7 Ia
9a 1 L
J L #$ m
=m
&
!
( 7R@CB
4.4.5. Integrals containing erf (z), erfi (z) and inverse trigonometric functions 1.
2.
3.
1
Z
0f
m
1
Z
1
Z
2
3 ¢ ¤
0f m W W Y Vf f9 - m , f9 #f9 3 ¢ - m , 3 1) )
W W
W m m ¢ - 0f9 ) , f9 1 ) ,
f m f <3 £ L<
fT3
C
¤
¤
+
4.
5.
6.
1
Z
1
2
1
8.
Z
W W -
1
m Vf 3
V -
W
3 f9 P3
f9 O £
f
3 £H¤
C
W m m ¢ ) , 3 f9 1 ) , f9 P3
¤
W m m ¢ ) , 3hf9 #f9 1 ) , 3
f9 C £ fT3 43f9
0 ¤ 0f
m
¢
f
Z
f
m
1
7.
f
Z
D£ m
Z
W W
5) 5) :
W W
W
¤
0f
D £ - W f9 3 m
3Pf9
£
£H¤
f C3
4.4.6. Integrals containing products of erf (z), erfc (z) and erfi (z) 1.
Z
2.
Z
3.
Z
4.
Z
5.
Z
7
7
f
0f
m 5 5( m 3 7 m m m
7 m m D F G 3 F 3 G9H
=m =m
=m
7 W 3 f 3 ¢ G / E / fg3E F h
=m
@CB
¤ 7 W / fg3E F - f 3 ¢ G
/ E
@CB
L L
@ B
@ B
6.
7.
8.
Z
# 0
/ T f 3E 9f
7 Y D 0f <3 0f f L 0f <3 0f L f H m 1 1 1 =m Z m W W £ 3 E
f - 1 ) , =m / E / fg
Z
/ E
-21 0fg3 -21
@ B
fg3
W W #$ # 7 5 ;5
7 7 1 1 1! f 4a F G N & & 9a a 1 9 a 1 5_7
m
( 7 @ B = ! W £ f W m m @ B = / E / fg3E
, 1)
fg3
@CB
9.
Z
Z
10.
W
/ E / fg3E
W
¢ 0f
3
=m
@ B
4.5. The Fresnel Integrals S(z) and C(z) 4.5.1. Integrals containing S(z) and algebraic functions 1.
2.
Z
-21 0fg3 -21
& & m
*
4a 1 0fg3 " "
Z
0fg3
F
G &N
W W $# & 5 & I5 & ! ( 1 & Ia a & 9a a
=m
!
( @ B
fg3
W 5 5 11L ( #$ & 1 1 f 4 a &4 J 5 N & !5 & 1& 1& L J
m = !
L P
( 7 7 C@ B
+
5) ; ,
4.5.2. Integrals containing S(z) and trigonometric functions
1.
2.
Z
Z
O' ! f
Y
3.
4.
SZ
0f f
7
5.
Z
6.
Z
"
W ' 5 " '#" m ' & #' 5 & ( #$ 1 ¤ O ' 5 7! % ' 5 ! N !
& ' 5 & V ' 5 # O' 5 7
" ]" 65 L J
- a L a a L J W J & a a ( #$ ! = ( X @ B N & & - a a a
0f f
0f £ £ f
m
SZ
m
"
£ 1
£ ) , &N
"( 7E( m
f f
7
W #$ 1 & # 7 ! ( ¤
& V7 ( # 7,5
W #$ 1 & 7 ( ! 7 5 ¤
( E 7 &
W #$ 1 & #7 (
! & #7 ( #7,5
m @CB =
1
, &N )
1
m ) , &N
"
& ( L J
&- - &- a ( (* ( n 5 & Y F G F G & N &- - ! ( &
=
P N
W
#$
@CB
# 0
4.5.3. Integrals containing S(z) and the logarithmic function 1.
Z
-21
Y
m 5 m (
£ '
7 f 4 a 4& F G 4&
W W Ia & 9a & ( #$
& ! 3 N & & 9a &
N &
9a & L
5 J ! 9a L JW W 9a & ( #$ ! 9a Ia ( @CB =m !
4.5.4. Integrals containing C(z) and algebraic functions 1.
2.
Z
-21 0fg3 -21 fg3 f 4a -21 F 7 *
4a 1 0fg3
Z
7
G &N
1 5 1 ;5 1 9a a 1
m = !
0fg3
5 L £ - -!&4 f9 4a 1 J 5 1 1 L N & J
W W #$ 1! ( 9a 1 a
&
( 7 @ B
W 1 5 ( #$ ! 1 ] 5 1 1
m = !
( C@ B
4.5.5. Integrals containing C(z) and trigonometric functions 1.
2.
3.
Z
' !
0f
'! J %
L
' &
W ' 5 1 ( #$ g '65 7! 1 N '65 ! '65_ 7 ¤ " 65 & !
- a L a a L * J W J a & a ( #$
! = ( X@ B a a a -
0f f £ 1
0 f Y f £ & N 1
SZ 7 0f 0f
W #$ # 7 ( £ m 1 ! f £ F G 1 N ¤ ( 5 m # 7 # < 7
& f
P1
Z
+
4.
SZ
"
"
5.
Z
6.
Z
Y F
5) ; :
W #$ 1 # 7 ! ( ¤ N
& V7 ( # 7< 5
W #$
1 #7 ! ( 1 N = m @CB
& #7 ( V7< 5
" " 1 ( L #$ J 1 1;- - 1;- a W G &N 1 1;- &- ! (
= 7R@CB
7 m * ¢ 3 V - S
7 m F G
7 (* (*
G F
7 (* 5n
4.5.6. Integrals containing C(z) and the logarithmic function 1.
Z
Y
-21
m 5 m (
£ ¢ N &
1 9a 1
m & 9 a 1 L
5 7X! J 9a & L * _ J W W #$ W W #$ Ia 9 a (
1 1! ( 1! 3 N & 9a Ia 9a & 1 &
m h7 @CB = !
4.6. The Incomplete Gamma Function γ(ν, z) 4.6.1. Integrals containing γ(ν, z) and algebraic functions 1.
& & " -21 0fg3 -21 fg3 4 m ' ; W #$ 5 I5 ( ! Y N = m 5 ! I5 ! & & 5_7 9a 5 9a a 1 5
Z 4a 1 0fg3 fg3
] 5 5n ! ( #$ £ - - -21 f9 4a a &4 5 5: O! N 5 7 5 5
5 5 L _ J
=m 5 ! Z
2.
@CB
P,
@CB
4.6.2. Integrals containing γ(ν, z) and the exponential function 1.
-21 0fg3 -21 + - . fg3 4
W #$ & & " 7 5 ;5 m ! ' ; & N & 5_7 Ia 5 9a a 5 1 m= 5 ! I5 ! @CB W 7 m Z ! . + - 0f6 3 4
, 5 1 L -21 ) J
= m @CB Z
2.
3.
4a 1 0fg3 / + - . fg3
Z
7 5 5 ! 5: ! #$ £ - - -21 f9 4a a 4& 5 n 5 5 L N 5_7 5 5 J
= m 5 ! ( A7 @ B ! Z . 3
-21 / + - 0fg ! 7 m G = m ! ( 7 X @ B F
5 1L * J
4.
4.6.3. Integrals containing γ(ν, z) and trigonometric functions 1.
2.
SZ
Z
9m
0f f
-
m
f f
W
£ 1 #7 ! ( £ & N & 5 7 #7 ( V<7 5
" " " 65: 5 7 ! a a a L a - a L #$
7 1 a J 5 5 5 7 J ! Y N =
_ 5 7 5 _ 5 7 a 5 _ 5 7 & & -
P 3
f
@CB
=
65 5_7!
@CB
+
3.
Z
0f f
W
5.
SZ
-
f f
a W
m
6.
-
-
-
8.
Z
Z
-
#-
-
a W
-
¢ 3
7.
#$ 5_7 ! 5 7 a 5 _
5 5? = 6
7X! O!
=
#$
@CB
@ B
=
#$ 1 # 7 ! = &m & N & 5 7 #7 ( #7,5
#$ 1 #7 !
&m & N & 5_7 #7 ( #7,5
= m
-S
@ B
1 #7 # 7 ! & m & N & _ 5 7 # 7 ( # 7<5
-S
0f £ £ f
£ 1 V7#7 ! £ & N & 5 7 # 7 ( V7<5
f f
W
¢ 3
SZ
5_7 L
-
9m
4.
" ]" 65n 5_7X! a - 5_7 L a a J J
7 a 1 5 5 Y N & & 5_7 - 5 5 7
SZ
5) :1 3
@ B
@CB
N &m & & 5 7 P95
1 #7 # 7 ! # 7 ( # 7,5
#$
= m
@CB
γ (ν, z )
4.6.4. Integrals containing γ(ν, z) and the logarithmic function 1.
Z
m 5 m (
-21
Y 2.
m 5
-21
Z
&N &
Y
&N &
m & 5 ! 5 5 ( m ! 5_7 5 5 1 5 5_7
m m (
5 7 5 5 m ! 5_7 5 5 1 5 5_7
5 !
5 5 1L
J
=m
5 !
5 !
@CB
& 5 !
!
J
=m
5
5 1L
@CB
4.6.5. Integrals containing γ(ν, z), erf (z) and erfi (z) 1.
2.
Z
"
Z
]0fg3
!
£ 0f O a &4 - £ f 3 £ )3 ¢ F - f GIH D 5 & J L =m
] fg3
Z
] fT3
#$
! & ! m a a 5 L f 4& 1 1 N 1 5 J
=m
3.
W W 7 ! ! a a 5 L f 4& 1 1 N a a J
#$
@ B
=m
@CB
@CB
4.6.6. Integrals containing products of γ(ν, z) 1.
Z
]0fg3 4 Y
1N
! ! 65 5n ! f a a 1 a W W #$ 7 ! =m a 5_7 a a &
P;
a
@CB
+
5) <
4.7. The Bessel Function Jν (z) 4.7.1. Integrals containing Jν (z) and algebraic functions 1.
2.
('%& !&
Z
5 ! ( & !&
&(' & ¢ ' 5 1 L 43 J $ L
5 m ( 7! ( 1 ( J L J Z
S
'
&\ - & - \ " #B 5: m !
. 0fg3 £ - !- & -21 / f9 4a + a & W #$ 5 5 5n L : 5 ( ! Y 5 J 7! 5 a L 1 N 5_7 5 a J
m= ! 5 ! ( 7A@ B . - 0fg3 - + a 0f63
m £ + a & . / f+ 1A- . -21 H G = m @ B F # " -21
. 0fg 3 + - 0fg 3
£ + &- . / f+ a 1 . -21 a F #" m G = m ( X@ B ! 1
m 3 !- &4 0fg 3 £ 1 F " G -21 0fg = m ! ( 7R@CB
0fg3 4a 1
3.
Z
4.
Z
5.
Z
6.
Z
7.
1
Z
-21
0fg3
7
£ £ D / 0f 3
0fg3
m F#" G £ / f
P :
m F #" G
m F#" GIH
=m =m
@CB
@ B
8.
9.
Z
-21 0fg3 1
Z
-21 0fg3 1 1
-!&4 fT3 -21 1
10.
m
G9H
=m
m F #" GIH
=m
1 F #"
@CB
0fg3
D / £
Z
0fg3
m D F #" m G 3
m F#" G 3 / f
fg3 m D ¢ 3
@ B
m F #" G9H =m
@CB
4.7.2. Integrals containing Jν (z) and the exponential function
SEa + &-21 . & + &-21 . V- F G
43 ¢ S £ -
Z
1.
4.7.3. Integrals containing Jν (z) and trigonometric functions 1.
Z
Y 2.
Z
/ E 43 ¢ SEa £ ES a 1 O/ S Y f + SEa a & . 3 ¢ $ L 43 3 ; S 3 m , - ) J .
] - + SEa a a & &SEa a a &4 f2 = m
SEa 0fg3
0f f
/ fg3
" ]" 5_7X! a - J
a Y N
& 5 7
65 5 7! 5_7 L a a J
a 1 a 5_7 !
a - 5_
7 a
P<
(
5_7 L W a
#$
5 7
f £ £ f
@CB
= 65 !
( 7A@ B
+
Z
3.
" ]" " 65 5_7X! a - 5_7 L J
J a a 1 a 5 7 ( Y N
7 a !
& 5 7 a - . 5_
f
Z
4.
£
-
5) < 3
0f
5_7X! a a W
#$
a 5_7
5_7 L = 65 !
( 7A@ B
W #$ " ']" " m '%&( 6 ' 5 ( 1 ' % '65 _ ' 5 5_ 7 ! '6 5_7 ¤ 5 7! 1 N 6
5.
SZ
-
-
6.
7.
Z
#-
-
£
Z
f9 3
3 ¢ SEa '
Y 8.
m SEa )
L J 5_7X! m ¢ 3 V - S
Z
W #$ 1 #7 ( ¤ ! N 5 7 # 7 ( # 7<5
& _
W $ # 1 #7 ! (
N & 5_7 # 7 ( # 7<5
= m @CB
F ¢ 3 J ( m ( %
0f £ f
L J _ 5 7X! m
'
f 3
G
G $ L &- F fT3 f 3 ;- m 5 m ( , S ;- a ) , =
F 43 ¢ E S a £ 3
P L
m @ B
f £ f G
¢ ' $ L]f &- E S a If S -; a O ¤ J
4.7.4. Integrals containing Jν (z) and the logarithmic function
1
1.
Z
2.
Z
7 f m fI <3 ¢ ¤
7
0 f O X £ f O <3 0f O £ 0f O X f O <3 f O = m m & Ia 5 ( Z L m L -21 m # 5 ! 9J a a J 5_7X! 1 L J W W 9a 9a ( #$ Y N 9a a I a ! 5_7 5_7 =m 5 !
& 1
Z m 5 m ( 1 C f O 3 0f O =m
1 Z m 5 m ( 7 ¢ 3 0f O
=m Z m 5 m (
& 7 f f O 3hf 0f O 3 = m ( m
f O
f
@CB
3.
4.
5.
6.
Z
7.
SEa -
0f
43 ¢ E S a £ fI S P < -21
)
P5
5 ] 5 ! 3
C,
=
@ B
@ B
@ B
@ B
W
m @CB
4.7.5. Integrals containing Jν (z) and inverse trigonometric functions 1.
1 -21
Z
f
9a a 1 L L 5 !J 5_J 7! 9a L N & J
P P
W 9a 9a a 1 ( #$ ! 9a 5_7 9a 5_7 5_7
= 5 !
@CB
+
1
2.
Z
3.
Z
4.
Z
5.
Z
6.
Z
7.
Z
1
1
1
2
1
1 7
5) < :
7 f m 0 fI ¤ f m F m G F m G ¤ 1 7 f m £ 1 f m D ¢ 3
fg3hf
fg3 0 fI ¤
F m GIH ¤
7 1 f m 0 fI 3
f ¤
7 f 3 ¢ ¤ 1 f 0 fI m g
4.7.6. Integrals containing Jν (z), Si (z) and ci (z) 1.
Z
-21 £ £ 0
" " 5 ! 7 3 3 G F ( _ 5 X 7 !
7( ( 7 ( 5
Y D F G F G 3 F ¢ 3
2.
Z
-21 £ 3 £ 0
" " 5 ! 7 3 F 3 G ( _ 5 7 !
5 7( 5 Y D F ¢ 3 G G F¢ 3 F
,9N N
5 F G
(
5 G 3 F IG H = 5 !
@CB
5 F G ( ,7 5 5 G 3 F 9G H (= 7 5 ! X@ B
4.7.7. Integrals containing products of Jν (z) 1.
Z
3.
4.
5.
6.
fT3
a g5 5_7 & N 1
m m
a a 5 5n & N g & a
Y
2.
65 m
Z
Z
7
7 m ( !
m
a 1 a a a a & 6 5
& a 5_7
5n 65
7 7 0fg3 D )(:7
1 a a & 5 5 1 6 a a ! 5_7 g 5 5 =m !
7 a 2- 1 fI g5 7 =m !
0fg3
5 5_7 g (87 7 D 65 (8 7 a 2- 1 fI 5_7 9m
7 m (
Z
a fI <3
m ! (& 65 5 7!
!
&
( 7R@CB
a a 1 fI H 7 ( 7A@ B !
a a 1 0fI H =m
fT3
7X!9 g5 (* ! 7 D 65 (: 8 ( 7!& (:7X! a 2- 1 0fI 9m g5 5_7X!9 65 5n O! g5_7X!9 5_7X! a a 1 fI H = m
( m #$ 5 7 ( m #$ !
!
7A@ B
!
. ,+ &-21 0fg3 & -21 / E / fg3
£ &-21 F 5 m G + a a 1 . a a F f f G
1
m= !
7R@CB
Z
Z
SEa fg3 a
/ E / fg3 3 ¢ SEa £ SEa . S ' 3 T3 Y F 5 m G + a a2SEa a 1 L $ L 3 8 3 S J J
,9N1
( 7R@CB
a 1 &-
+
" . Y & " F 5 m G + a
7.
Z
8.
Z
(
7
1 3 F
Z
1
10.
Z
11.
Z
7 ( m 3 F 1
12.
a a a a2SEa a 1 F f f G = m ! ( (87 ! ( ')(:7R@CB
&-21 0fg3 &-21 / E 1 -; / E / fT3
'%& . 43 ¢ &-21 m G + a a 1 & -21 ' (87 7 G
L
3hf G
3 H f
D 0f O ¢ £ / f I KD ¢ L £ / f I H 3 £ / f I £ / f I ¤ 1 1
Y
F
,9N,
5:7! 5n7 ' 5 1 ( m ¤ O' 5 5 5n7 O ! O' 5n7 a
Z
13.
X- -
0f 0f
" '#" " m '%& &( '6565 5_7! 5_7 L 5g5 7 L 5 5_7 L ' ! J
J a a a 1 J a a 5_7 a 1 ( m N 565_ 7 5 5_ 7 565 5_!7 '65_7 ¤ 5 6 5 5 7 & L J
Y
Z
14.
0f
15.
1A- -
V-
- -
17.
18.
-
&(
&( 7 ( " ( J L 65_7X! 5 7! m
Z
16.
(& m ! L 65_J 7! 5_7X! 7 ( m
SZ
-
a a 1 a 5 7 #7 & ( ! N 65_7 5_7 g 5 _ 5 7 &- & a
#$
¤
a a 1 a 5_7 1 #7 ( #$ ¤ ! N g5 7 5 7 65 _ 5 7#7 ( #7,5
a a 1 a 5_7 1 #7 ( #$ L ! 65_J 7X ! 5 7 ! m N 65_7 5 7 6 = 5 5 7 #7 ( #7,5
Z m 5 m ( m £ f £ f O 3 £ f O
1 m £ £ £ £ 1 f O H f O <3 f O H1 f O Z m 5 m (
& 7 m £ f ] f9 3 £ £ f O 3 0f9 3 V £ f O 0 1 m £ £ £ £ Vf 3 1 f O H f O 3 f O H1 f O 0 ,9N 3
@CB
m
=m
=m
@ B
@CB
+
. 0f9 32
R+ &-21 & -21 F 43 ¢ m m Y ED F
Z
19.
1
20.
Z
21.
Z
22.
Z
1
1 7
&-21
m m F 3 G9H W = m ! ( 7R@CB 7
1 fI <3 m fI 1 fI H ¤
7 ¢ 3 0f9 ¢ 0fI f fI fI <3hf9 0fI ¤ m 1 1
1 f
£ &-21 / G
7 f 3 D fI
1 0f
3 7 fg3 £ f2 Vf9 ¢ £ fI Vf9 3 m m 1 £ If H
1 f/ -21 Z
23.
f 3 G
5) L1
3 ¢
f /
¢
/
3 ¢
/
¢ £ If 1 £ fI <3 £ If H £ If ¤ 1
¢
£ N 1
WW
#$
1! ¤ 7 #7 # 7 &
4.8. The Bessel Function Yν (z) 4.8.1. Integrals containing Yν (z) and algebraic functions 1.
7 ( m
Z
F m G F m G
@ B
=m
=m
4.8.2. Integrals containing Yν (z) and Jν (z) 1.
Z
7
1 0 fg3
0fI 0 fI m 1 ,9N95
@ B
2.
Z
D
-
H
1 F ¢ 3
m m 5 m 5 m 5 G
= m
@ B
4.9. The Modified Bessel Function Iν (z) 4.9.1. Integrals containing Iν (z) and algebraic functions 1.
Z
-21 0fg3 -21 0fg3
2.
Z
3.
Z
4.
Z
F
5 I5 m & &( " ! G 5_7X! F G N & 5_7
#$
¤
0fg3 fg3
/ *F m G a 1 F G 1 a F m G 1
m fg3 F G
m -21 0fg3 -21 0fg3 V F G
=m
!
( 7A@ B
=m
@ B
=m
@ B
. fg3 £ - !- & -21 / 9f 4a + a & W 5 5 5n L 5n #$ ! ( 7R@CB ! Y 5_J 7! 5 5 m = !
a L 1 N 5_7] 5 a J
Z . 0fg3 + - 0fg 3
6. £ + &- . / f + a 1 . -21 a F #" m G = m ( @ B ! 1
5.
Z
0fg3 4a 1
,9N;
+
7.
8.
9.
5) P1 ,
. - 0fg3 - + a 0fg3
m £ + a & . / f+ 1A- . -21 L G F # " -21
Z
Z
-21 0fg3 -!&4
Z
1
7
fg3
10.
Z
-21
11.
Z
-21 fT3 1
Z
-!&4 fT3 -21 1
13.
m F#" GIH
( @CB
( @CB
=m
0fg3
m D F#" m G F #" m 9G H 1
=m
=m
@CB
m F #" G
@ B
@CB
Z
!
-21 fT3 1 1 0fg3
MD / f F # " m G 3 / £
12.
=m
0fg3
m fg3 £ 1 F#" G =m !
m £ £ £ D / f F#" G 3 / f
=m
0fg3 m D
m F #" GIH
@ B
m F #" G 3 ¢ H
=m
@CB
4.9.2. Integrals containing Iν (z) and the exponential function 1.
Z
-21 0fg3 -21 + - . fg3 4
W & & " 5 1 5 ;5 m 5_7X! F G & N & 5_7 Ia 5 9a ! a 5 1
5 ! I5
! =m #$
,9N :
@ B
2.
Z
3.
Z
4.
Z
6.
7.
8.
9.
10.
@ B
=m
=m
@ B
. + - 0f63 4
5.
m . + - fg3 4 " ) f * ,
2
. + - 0f63 4 " ) f * ,
m 5_7
f , 3 m ) * *
. -21 + - 0f63 4
m m m W , 3 a 1 ) , * -21 ) W 7 Z . 3 4 m £ 3h9f K3 £ -21 + - 1 0f6
W
@ B
=m
Z
=m
=m
@CB
. -21 0fg3 -21 + - fg3 4
m m 7 W =m * , 3 a 1 ) , 2 - 1 ) W Z . 3 -21 + - 1 fg 3 4 m £ 3hf9 K3 £ -21 0fg =m
@ B
Z
. 1 / + - 0fg3
m D 3 F 7 h 3 f G " m / f H
@CB
@CB
Z
Z
. -21 / + - 0f63 " / f ,9N<
=m =m
@ B
@ B
+
5) P1 3
. -21 / + - 1 0f63
7 £ 3 / f / f K 3 £ m
Z
11.
. -21 fT3 -21 / + - 1 fg3
m
=m
@ B
Z
12.
3hf 3 ¢
=m
@CB
4.9.3. Integrals containing Iν (z) and trigonometric functions
Z
" ]" " 65 5_7X! 5_7X! a - 5_7 L a a 5_7 L J J W #$ a a 1 a 5 7
65 ! ( 7A@ B ! Y N =
5 7 a _ 5 7 a a 5 7
& -
Z 0f
2.
f " " 65 5 7! 5_7X! a - 5_7 L a a 5_7 L J
J W #$ a a 1 a 5_7
£ ! Y f £ N = 65 ! ( 7R@CB 5_7 a - 5 7 a a 5_7 0f &
SZ 0f
3. -
0f W #$ 1 #7 £ m L f ! f £ J 5_7! N & 5_7# 7( #7<5 ¤ m
SZ £ m f £ f 4. -
m 0f f L 1 1 & #7 Y J 5_ 7! N 5 7 #7 ( V 7 ( #! 7,5 V7<5 ¤ 1.
f
,9N L
5.
SZ
-
-
¢ 3
SZ
6.
-
W a
-
Z
8.
Z
-
-
-
W a
-
f 0f " 5_7X! a 5_7 L - 5 7L J
J
Z
10.
Z
f f
#$
= m
@CB
0f £ £ f
1 #7 !
#7<5
N
#$ 1 & a 1 #7,5 ! 1 1 # 7,5 - # 7,5 a
= ( R7 @CB
" " " '& 65 5n O! 5_7X! a - a &L a a a &L J
a J a 5_ 7 a a #$ a £
& & ! 0f £ N f & 5_7 5 & a - a & a a a &
= g5 !
7 5 1 #$ L 1 J 5:7! m & N & 5n 7 7( ! 725 ¤
W #$ # 7 L 1 ! J 5 7! m N & 5_7# 7( V7<5
= m @CB
5 1 L J 5 7 ! m & N & 5_7 #7 (
9.
W #$ # 7 L 1 ¤ ! 5 7X! m N & 5_7V 7 ( # 7<5 J _
¢ 3 # - S 7.
-S
Y
,9N P
( @CB
+
5) P1 5
f
f " ]" 65 5 7! £ 0f a a 5_7 L f £ 5_7X! a - 5_7 L a & J a a 5_J 7 a a & #$
65 ! ( @CB ! Y N =
5_7 5 a a a a a - & & & &
Z
f
" " " '& 65 5n O! 5_7X! a - a &L a a a &L J J a & a a 5_7 a a & #$
! Y N = 65 ! ( C@ B & 5_7 5 & a - a & a a a &
Z
f
" ]" " 65 5_7X! 5_7X! a - 5_7 L a a 5_7 L J J
#$
a 1 a & a a 1 a 5_7 ! Y N = g5 ! ( C@ B 1 5 1 5_ 7 a - 5_7 a a 5_7
" ]" Z 5_7X! ( !
W #$ a (65 ()( - ! Y F G F G N & 5_ 7
a - 1
= ( ! @CB
Z
11.
12.
13.
14.
4.9.4. Integrals containing Iν (z) and the logarithmic function 1.
Z
-21
m 5 m ( Y
9a
N & 9 a a
9a L m & L # 5 ! 9J a a 1 JL 5_7! WJ W 9a #$ ! =m 5 ! 1 Ia 5_7 5_7
,N
@ B
m 5 m (
2.
Z
3.
Z
4.
Z
&
1
Z
6.
Z
7 1 f O <3
m 5 m (
f O <3
C
m !(:7
@ B
=m
=m
7 m 5 m ( f # f O 3 f 0f O 3 = m 7,5 7 ( m ¤
7( 7 ( f
m " " m &( 5 ! -21 m 5 m ( 5 ! 5 5 1 L 5_7! J
5 1 5 5 ! m #$ Y N 5 7 5 5 5 5_7 =m 5 ! 1 & &
5.
@ B
@ B
@ B
4.9.5. Integrals containing Iν (z) and inverse trigonometric functions 1.
1 -21
Z
0f
9a a 1 L L 5 !J 5_J 7! 9a L N & J
1
2.
Z
3.
Z
4.
Z
5.
Z
1
1
7 0f m
2
1 7
W #$ Ia 9a a 1 ! 9 a 5_7 9a 5_7 5_7
= 5 !
0fI ¤
m m f m F G 1 F G ¤ 7 0f m f 1 0 f
fg3 £
0fI
7 ( ,
m
m ¤
f-
0fI ¤
@CB
+
1
6.
Z
7.
Z
1
5) P1 :
m 1 0 f m D F G 3 ¢ H ¤ 7 1 f m fT3
fI 0 ¤
4.9.6. Integrals containing Iν (z) and special functions Z
4 3 £
" 5 5 5 5 ! 1 3 5 5 7 O 5n7 7 (1 L C 7258 ! & N 5 : ! J
( ! =
1.
-21
2.
£ ] fg3
f & O 1 2- 1 0f O 1 0f O 3 -!&4 0f O 4& 0f O
3.
Z
4.
Z
5.
Z
6.
Z
£ ] fg3
£ ]0fg3
fg3
SZ
Y
@CB
0f f
m * 3
-
/
f £ £ f
,!,
@CB
/ £f¤
m / £ f K3 * ¤
=m
/
J L _ 5 7X! m N
"
&(!& ( & & m 65 : 5 O!
/ E / fg3E
= m
£
!
( 7R@CB
m
#$ 1 V7 ! ( ¤ a 1 5_7 5_7V 7 ( #7,5
7.
SZ
-
-
¢ 3 Z
8.
#-
-
/
-S
/
/
L J _ 5 7! m N
/
#$ 1 #7 ! ( ¤ a 1 5 7 5 7 #7 ( # 7,5
#7 ( L 1 J _ 5 7! m N a 5 7 5_! 7#7 ( #7,5 1
Z m 5 m ( 9. 2- 1
" " m & Ia L 5n ! 9a a L J 5_7X! J #$ 9a 9a ( Y N 9a a 9 a 5_7 ! a 5_7 5_7
1
Z1 F f ¢ 3 G / ¢ 3 / ¢ £ 10. -21
#$
= m
@CB
5 ! = n
@CB
W #$ 1! ¤ 1 N & 7 #7 &
4.9.7. Integrals containing products of Iν (z) 1.
2.
3.
Z
fg3 ]0fg3 4
7 7 F G a a 1 f O = m ! ( 7 @CB Z m V 0f63 f 0f O f O L1 f O <3 1 0f O L 0 f O = m @CB Z m m 0f O! # fQ3
f O L1 f O 3 1 0f O L 0f O = m @CB f a a 1 *
,3
+
4.
5.
6.
7.
8.
9.
10.
5) P1 <
V1 0f63 1 0 f O 3hf f O m 3 0 f O L1 f O 3 1 f O L f O 0 = m Z -21 V 0fg3 £ f 1 / £ f £ £ £ £ f 1 / f L1 / f <3 1 / f L / f = m Z £ £ 3
3 f 1 / f 1 / £ f -21 1 0fg £ £ £ £ f 1 1 / f L / f <3 / f L1 / f = m Z
Z
-21 0fg3 1 fg3 m / £ f L 1
Z
-!&4 0fg3 -21
Z
-21 0fg3 -21 V 1
Z
1 fg3
/ £ f <3 / £ f L1 / £ f
@CB
@CB
fg3 1
=m
0fg3
/ £ f <3 ¢ m £ / fg3
m 1 f <3 m
@CB
=m =m
@CB
@ B
] fg3 4
5 1L 5 1 m #$ F m G a 1 ! 5_7J ! 5 & L 1 N 5_7 5 & J
=m !
&0fg3
@ B
11.
12.
Z
] fg3 4
1
( 7 @CB
D £ f O £ f O L1 £ f O 3 1 £ f O L £ f O H ¤ 7 7 # 0f V1 0fI 3 V fI m fI 1 fI ¤ f
Z
,5
13.
14.
15.
7 1 0f m ¢ f 3 ¢ 0fI 3hf fI 1 fI 3hf
Z m 5 m ( m
£ f £ f O <3 1 £ f O 0 m £ £ £ £ f O L1 f O <3 1 f O L f O Z m 5 m ( #
& 7X m £ f ]0f £ £ f O 3 0f V £ f O 0 1 m £ £ £ Vf9
f O L 1 f O 3 1 f O L £ f O 0 1
Z
. ,+ -21 fg3 £ / E V
'& 5 m 1 L J
5_7X! 5 & L N J
16.
I ν (z )
Z
1 0fI ¤
=m
@ B
=m
@CB
/ fT3E
W #$ 1 5 1! = m @ B a 1 5_ 7 a & 1 a 5_7
" " m !& 5 Z 1L 5_7X! J 5
0fg3 / E / fT3E
&L W J
#$ 5 1 1 ! Y N = m @ B
a 1 5_ 7 5_ 7 1 5_7 5 &
Z1 f / ¢ / ¢ f / ¢ 3 / ¢ 3
W W #$ -21 1! ¤ £ N 1 7 #7 #7 &
17.
18.
19.
Z
- fT3 -
/ E - / E / fT3E 1 5 ! C7 ( ! F G -21 N
7( & ( & ( L m J
,!;
f 3E
- / g W #$ 1! a 1 1 5_7 5_7
=m
@CB
# #
5) N1
4.10. The Macdonald Function Kν (z) 4.10.1. Integrals containing Kν (z), Jν (z), Yν (z) and Iν (z) 1.
Z
2.
Z
3.
7
1 fg3 m 0 fI 3 1 0fI ¤
7
-
7 m D 3 1 H F m ( 3 ¢ G 7 1 f 1 f 3 m
Z
m 5 m (
m ¢
3 , )
= m
m = ?
@CB @CB
4.10.2. Integrals containing products of Kν (z) 1.
"
Z
5 m ! If
£ - V/ (:7X! f - 2- 1 T3 ¢ 0 fI £ T 3 ¢ f 1 fI 0 5 1 !
= m 7A@CB 7 7 Z F G 3 = @ B &
%
2.
= @ B
3.
Z
4.
Z
&
3 7
5.
Z
3 7
6.
Z
7.
Z
= @ B
3 7
&
1
= @ B
7 7
= @ B ,:
= @ B
L
H
2 % & 1 7
= @ B
Z
8.
% & 1 3 7 Z 7 10. % 1& 7 7 Z X 7 11.
3 1
Z
9.
= @ B
= @ B = @ B
4.11. The Struve Functions Hν (z) and Lν (z) 4.11.1. Integrals containing Hν (z), Lν (z) and algebraic functions 1.
Z
-21 0fg3 -21
£ - f 4a a
Y
2.
Z
- 0fg3 -
3.
Z
4.
Z
L
0fg3
L
&
&N
fg3
0fg3 5 a 1 ! I5 a 1 ! a 1 5 & L 5 I5 5_7! J W W 7 5 a 1 I5 a 1 #$ ! 5 & 9 a a a 1 Ia a 1 5_7
=m 5 ! 5 ! !
L
H
m £ / f 1 - -21 *G 3 F -21
m 0fg 3 D F *G 3 ¢ H
L
H
5 7 O! F G 2- 1
0fg3 £
fg3 7] 5 5 a L 5 J 5 L 5 &L N & & 5 J J
4a 1 0fg3 Y
,!<
( 7A@ B
=m
@CB
=m
@CB
.
-!& + a 1
f9 4a a a 1 W #$ a !
& 5 5
m 5 ! ( X@ B = !
# #
5.
Z
-21 H
6.
Z
-21 0fg3 -21 H
7.
Z
8.
0fg3
D¢ 3
0fg3
5)
m F #" GIH
m
m F" G
-21 0fg3 1 H1 0fg3
¢ 3 F #" m G 3 m -!&4 0fg3 -21
H
1
Z
m m F#" G!
0fg3 3 m
11.
12.
Z
13.
-21 fT3 -21 L
fg3
-21 fT3 1 L1 fg3
¢ 3 F#" m G m
@CB
=m
m
=m
m m F#" G! 3
=m
m F #" G
@CB @CB
Z
-!&4 fT3 -21 L1
Z
@CB
=m
=m
fg3 m
-!&4 fT3 - L1 fg3
D £ F #" m G 3 / £ f F #" m G f
m
14.
@CB
Z
m F#" G £ &4
= m @CB
-!&4 0fg3 - H1 0fg3
DC3 £ F #" m G / £ f F# " m G f F #" m G9H m Z m ¢ D F #" G 3 H 10. 3
-21 L fg 9.
=m
Z
@CB
=m
@CB
m F #" G 3 £ &4
= m @CB
,L
m F #" 9G H
=m
@CB
L
H
4.11.2. Integrals containing Hν (z) and hyperbolic functions 1.
Z
( m
/ fT3
H
/ E
W '& !& 1 7 # m ! 5 L N & & a a J
#$
=m
!
( @CB
4.11.3. Integrals containing Hν (z), Lν (z) and trigonometric functions
Z
1.
- 2- 1
£
2.
Z
- 2- 1
0f f
0f
' 5 1 '65_7 6 " '#" " m ' & '& N ' 5 5 & '65 'g5 & L 'g5 5 & L & 6 J J
f
H
H
4.
Z
0f f
5: O!
L J 5 &L
W #$ a a & ( ! a a a & 5 &
= g5 ! K(
" " " !& 65 : 5 O! a - a &L a a a &L 5 &L J J ,P
J
L
W #$ ( ¤ ! & O'65_7
W #$ ' a 1 5_7 ! ( ¤ N & a 5 a ]6 & & ' 5_7
" " " !& 65 a - a &L a a a & J J
7 a 5_7 £ f £ N 0f & & a - a &
Y
Z
" '#" " m '%&(!& a & a & 5 L J L J
3.
H
7X! @CB
# #
Y
5.
SZ
- 2- 1
f
7 a
f £ £ N 0f &
&
H L
a
5_7 a a & ! - a & a a a &
= g5
" " '& ¢ 3 fI 5 & L m J
SZ H f
- -21 L " " '& fI 5 & m J L SZ H
- - -21 L " " !& ¢ 3 - S 5 & m J L Z H
- - -21 L " " !& 5 ! N m & & 5
&N
7.
8.
&N
&N
W #$
5 & ! (K
W #$ 1 V7#7 ! ¤
& 5 & #7( V7<5
W #$ 1 #7 #7 ¤ !
& 5 & #7 ( # 7,5
W 1 #7 #7
& #7 ( ! # 7,5
#$
= m
4.11.4. Integrals containing Hν (z), Lν (z) and the logarithmic function 1.
2.
7X! @CB
W #$ 1 #7V7 ! ¤ & 5 & #7( #7<5
6.
5) 5
@ B
&(!& !& &a a 1 L m L H -21
5 5:7! J a 5:J 7 L 5 & L L J
J
W W #$ 7 a a a a 1 1 ! Y N = m ! 5 ! ?( A7 @ B
5 n 5 a & & 7 &a a & &
7 Z m 5 m ( f 3 f O 0
H
= m C@ B Z
m 5 m (
, ,9N
#
3.
Z
" "! #$ " #$ $!
m 5 m ( 7 f O 3 f X L
@CB
=m
4.11.5. Integrals containing Hν (z), Lν (z) and inverse trigonometric functions
1 -21
f 0f
W !& 7 9 9a 5_7 L a a 1 9a 5_7 #$ m ! 5 5_7! J 5 & L 9a a 1 L & N & & 5 & 9a a & 9a a & J J
5 ! ( R7 @CB = Z1 7 C 3 0fI f = m C@ B 2. H f
m 1.
Z
H L
1 7
3.
Z
4.
Z
5.
Z
6.
Z
1
H L
1 7
1 f
C3
7 f m 0fI 3 ¢ 3 1 0f
¢ 3
L
fg3
H
C
0fI
C3
f
m
7
f
f
fI <3
fg3 m
7
f
= m
@CB
= m
@CB
@CB
= m
/ E L / fg3E
W #$ & '& 1 #7 ! m &( g5 5 ! N !
& & 6 5 5 & 65 5n
= m !
( @CB
4.12. The Kelvin Functions berν (z), beiν (z), kerν (z) and keiν (z) 4.12.1. Integrals containing berν (z), beiν (z), kerν (z), keiν (z) and algebraic functions 1.
Z
7
m (
m D F G 3 , ,
m GIH F
=m
@CB
# #
7 ( m
2.
Z
3.
Z
4.
Z
F m G F m G
7 ( m
7 D
7 ( m
5) 31
m G 3
F *
m G F
m G9H * F
m G F
@CB
=m
=m
=m
@CB @CB
4.13. The Airy Functions Ai (z) and Bi (z) 4.13.1. Integrals containing products of Ai (z) and Bi (z) 1.
2.
3.
4.
5.
6.
7.
Z
-21
Z
-21
7X
3
! 1;- L J 1a L - L J & J & 7 @ VB 7 !C@CB = ! = L 9a 1 L 3 " 8 J I a L J - L J J = ! = @& #B !C@MB " 3 & 8 ! = @CB 9a L J 3 " 8 3 " 8
5_7 F G F G
@ & = = B O!C@MB
¢ £ 4 a . - +
Z
-21
" & 3 8
Z
-21
Z
-21
Z
V-
!
3
7(* .
43 ¢ £ - + 4 a T F G = 7 = @ VB !C@CB
#$
1L L 1 1 &]
7 J & 3 J & N 1 & ! 1
1 L L 1 1 3 J & J & N 1 & &]3
! 1
!
= = @ VB 7 7!C@MB
Z
, , ,
#$
= = @ B 7O!C@MB
& '
#$
1L L 1 1
8. X7 J & J & N 1 & 1 ! Z
L 1L 1 1 43 3
9. J & 3 J & N
1 ! &1 #$ Z
L 1 L 1 1 10. 3 3
3 J & N 1 & J & ! 1 Z
= = @ VB C! @MB
#$
= = @ VB !C@MB
= = @ B !C@MB
11.
Z
&
12.
Z
]& 3
13.
Z
14.
Z
15.
Z
16.
Z
V
#
= = @ B !C@MB
3
7
7 3 7X
= = @ VB 7 !C@MB
7X 7
= = @ B !C@MB

= @ B
" 7 2 ;F G 3 7 F G
= @ B
7 G 7 F G 3 7X F
= @ B
4.14. The Legendre Polynomials Pn (z) 4.14.1. Integrals containing Pn (z) and algebraic functions 1.
1
Z
2.
Z
m 5_7 3hfI -21 j O'65_7
0f 3 -21 j
43 ¢ J 1 ' L % ' j F
, ,93
¢ 3hf G
= m
@CB
= m
@CB
# #
3.
4.
5.
1
7.
Z
8.
Z
11.
m (
! ' & j
m' ¤ 7 ( m
7 a '65_ 7 F m : ( 7 G 1
5 m ' & j a
1
( 7X! ' 6 ' 5 f - &- 0f9 ¢ -;&-!&4
m (87!
B
@ B
= m
= m
@ B
0f 3 -21 ¢ 3 - -;&-21 j
' 7 A. ( 7! F G f9 -21 ¢ 3hf9
-;&-21 j + -21 l ¢ 3 £ 9f O = m @CB
0f9 32
-21 ¢ 3 2
- ;- &-!&4 j a 1
' A. ( 7X! F G f 4a 1 ] ¢ h 3 f -;&-!&4 j + 4 a 1 l ¢ 3 £ f =m Z " ¢ O'65_7 ¢ 3hf9
-;&-21 j a 1 f O ¤ 3 2
-;&!- &4 j
1
Z
-21 Z1 -21
12.
7
-21 7 ( 7X! ' Z1
' & j
'65_7 f - &- f9 ¢ -;&-21
5 m
6.
10.
7(* m 5 m j
-21 Z1
Z
9.
7
1
Z
5) 5)
Z
7 (
7
¢ 3 j £
C
£ ¢ ¤
7 ¢ £ &4 _¤ C7 ( ! 3 j
j ¢ 0fg3 4
O'6m 5_7 ¢ m
*
, ,!5
¤
@CB
& '
13.
14.
15.
16.
0fg3 -21 ¢ - ;- &-21 # ! '! % '&
Z
j / ¢
f ¢ f O -;&-21 4 a 1 / ¢ f Z 0fg3 -21 ¢ -;&-!&4 j / ¢
'6m 5_7 ¢ f O -;&-21 / ¢ f
Z
-21 fT3 -21 j fg3
3 ¢ J 1' L % ' j ¢ 3 m
*
=m
@CB
@CB
=m
@CB
!
@CB
Z
-21 fT3 -!&4#j ¢ fg3
/ £ j ¢ m
, ) *
Z
j ¢ 3 0f63
m 4 3 ¢ ' % f -21 f / j £j 3 ,
&-21 ) ) & J L' 7 Z ¢ 3 fg3
20. 7( m ( ! j a 1 ' # ( 7! ' % & L ' J
, , ;
19.
=m
18.
17.
-21 fT3 -21 j ¢ fg3
" & m " ! ( ' ]'65_7 m 5 1L & N 7 5 1 ! ( = J
Z m m -21 j ¢ fg3 'g5 7 ) * ¢ , Z
@ B
=m
m ,
=m
m j a 1 ) ,
@CB
=m
!
'8>?7A@ B
=m
@CB
# #
Z
21.
43 ¢ Z
-21 j ) " ¢ ' (87! % a 1 1 J L' " 7( m (
5) 5) ,
3 0fg3 ,
m m m F G 1 * j ) * , 3 j a 1 ) * ,
" ¢ 3 0f63 ,
j a ! 1 ) 43 ¢ ' % j a
1 &L ' J
7,5 m ( ! Z 3 j ) 7( m ( ! ,
-21 ¢ 3 0fg " ' 3 ' % -21 ] f 3 £ a 1 j a
1 ) * & J L'
22.
23.
Z
-21 fT3 -!&4 j
24.
W
Z
-;&-!&#-R j F G
25.
=m
@CB
m , ) *
=m
m m (* ,
=m
@CB
@ B
0f63
¢ 3 m 43 ¢ / £ J 1' L % ' j
*
"' " m ' & V -R W F m G
=m
@CB
= m
@CB
4.14.2. Integrals containing Pn (z) and trigonometric functions 1.
Z
0f j
f " 65_7X! 1 L ' 3 ¢ ' % 5 7 L J a 5_7 L - _ J J
(' ]'65 1 a 1 Y N 5 7 & 1 - _
, ,9:
0f £ £ f 5_7 #$ ! a 5_7
=
( 7R@CB
& '
" " 65n O! & L ' 0 f ¢ j a 1
43 ' % 2. a J a & L f - a &L
#$ J
( ' ] '65 5_7 J a £ & & ! Y 0f £ N = ( X@ B & - a & a a & f &
'#" ' )(*'65_7! Z 1 0f G J L ' 5_7 j F 3. ' _ 5 7 a % f - - L - L J J
" # $ ( a _ 5 7 £ 1;- - 1 - ?')(:7A@CB ! Y 0f £ N =
* ( ' _ 5 7 a _ 5 7 & f - - - 1
(' '65 #$ m ! Z 1 & ¤ ¢ 4. f j 7 ( m &N
F G
&- & a ! (
SZ 0f j
5.
f ( ' ]'65 1 #7 ¤ 3 ¢ ' J 1% L ' m f £ £ N ! m f & 7 ( #7, 5
SZ 0f j
6.
f 3 ¢ m f £ £ N ( ' 7]( '65_V7<75 1 ! 7 ¤ m
m m f &
SZ 7 f j a 1
7.
f (' 'g5 #7 #$ & 1 ! 3 ¢ ' % F G m f £ £ N ¤
m V 7 ( # , 7 5 & f &
SZ 7 0f j a 1
8.
0f ( ' ]'65 & 1 7 m f £ £ N ¤ ! 7 ( # , 7 5
m
& f
Z
, , <
# #
9.
10.
11.
12.
13.
14.
15.
SZ
SZ
SZ
SZ
SZ
SZ
SZ
0f j ¢
f m m
f j F 0f m
7,5
7
¢
m
5) 5) ,
f £ £ f
(' '65_7 1 ( ¤ 7 ( #7,5 ! &N
G
f f
f j a
1 0f m m
F
( ' ]'65 1 1 ( £ ¤ £ &N
7 ( # <7 5 !
¢
f f
£ £
G
(' ]'65 & 1 ( ¤ 7( # 7<5 ! &N
7 f ¢ j ) 7,5 ,
0f ( 1- 1 ( m f £ £ N ! m f & 7( #7<5
f j F G
0f " ( 1;- 1 #7 ! £ m f £ £ J ' 1 % L ' £ O N m & 1 (*' V 7 ( #7,5 f
7 f j F G
0f ( 1;- 1 7 m f £ £ N ! m f & 7 ( #7,5
f j
0f '%& 1 L ' ( ' (' 1 #7 f £ m ' %J m f £ N & ( ' V7 ( #7,5 !
, ,9L
¤
#$
¤
¤
¤
& '
16.
SZ
j
-
3 ¢ J ' 1 % L ' ¢ 3 m
17.
18.
19.
20.
21.
22.
S Z "
j a 1
S Z "
7 j a 1 m ¢ 3
SZ
j ¢
V-
j F
SZ
-
¢
-S
7
m ¢ 3
&N
-S
G
7 ¢ 3 # - S m
SZ
7,5
"
j a 1 F
¢
-
¢
j )
7<5
7
7 ¢ 3 m , ,9P
( ' ] '65 1 1 ( #$ ¤ & N 7 ( # 7,5 !
G
7 ¢ 3 m
SZ
( ' ] '65 & 1 #7 #$ ¤ ! N & & #7( V7< 5
#$ (' '65 & 1 ! 7 ¤
7( #7, 5
( ' ]'65_7 ( #$ 1 ¤ ! 7 ( #7<5
&N
3 ¢ J & ' L% ' ¢ 3 # - S m
( ' ]'65 1 #7 #$ ¤ & N 7( #7, 5 !
-S
-S
( ' ] '65 & 1 ( #$ ¤ & N 7 ( # ,7 5 !
,
-S
( ;1 - 1 ( #$ ¤ & N 7 ( # 7,5 !
# #
23.
SZ
-
j F G
( ;1 - 1 V7 " #$ S N ! ¤ & 1 ( ' V 7 ( # 7<5
( #$ 7 7 ¢ 1;- 1 ! 7 ¤ S G
A j F m 3 - & N 7 ( #7,5
J ' 1 % L ' £ O ¢ 3 m
24.
25.
SZ
SZ
-
V-
j
£ ¢ 3 # - S
26.
Z
27.
Z
28.
Z
29.
Z
30.
Z
31.
Z
-
-
" "
#-
-
5) 5) ,
( ' ( ' 1 J ' % L m ' N ( ' V7 ( &
( ' 'g5 1 V7 #$ 1 ' L j 3 ¢ J ' % m & N 7 ( # 7< 5 !
(' '65 &L ' ¢ j a 1
43 J ' % m 5_7 & N
&-
j a 1 j a 1 j ¢ j F
( ' ]'65 3 ¢ J & ' L % ' N m & & V 7 (
&
#$
#$
¤
@CB
= m
#$ & # 7 !
& a
= m 1 V 7 #$ ! V7< 5
= m
( ' ]'65 & 1 #$ 7 !
m & N 7 ( #7, 5
(' '65_7 ( #$ 7 1 !
m & N 7 ( #7,5
(' ]'65 1 1 ! ( ¢ G 7 N m & 7 ( # 7<5
,93 N
1 # 7 ! #7<5
@CB
@CB @CB
= m
= m
= m
@CB @CB
32.
Z
& '
7<5
Z
33.
Z
34.
35.
Z
36.
Z
"
-
¢
j a 1 F
#-
j F 7
-
j F
-
j
¢
j )
G
( ' ] '65 & 1 ! ( 7 N m & 7 ( # 7,5
7 7<5 ,
#$
= m
( #$ ;1 - 1 ! ( 7 N = m m & 7 ( # 7,5
( 1 #7 " 1 ' L 1 £ J ' % m O N G
' #7 ( #7,! 5 & 1 (*
= m
#$ ( 7 1;- 1 ! 7 G
= m m & N 7 ( #7<5
(' V(' V7 #$ 1L' 1 £ J ' % m N & ( O' #7( #7,5 !
= m
@CB
@CB
#$
@CB
@CB
@CB
4.14.3. Integrals containing Pn (z) and the logarithmic function
; $ # $ f Z
1.
fg3 &-21
-
2.
Z
¡=
m 5 m (
( L 5 m J ' G j F m G
F m T j ¢
7
', 'g5 7! 3.
Z
m 5 m (
'
' % £ 3 ¤ 1 J L'
'65_7! % . ¢ ¢ j + -21 lC-!&4 f O <3 ¤ 1L'& a 1 J
j / ¢ 3 E
7 '65_7X!A O')(87! ¢ 3 '65_7X! % j + -1 lC- . ¢ 3 £ f O ¤ a 1L'& 1 J
,931
# #
4.
Z
m 5 m (
5) 5) 5
' & ( 7 ( !
7 ( '65_7X!A 7 O'65 7!
¢ 3 '65_7X! % j + -1 lC-21 . ¢ 3 £ f O ¤ a 1L'& 1 J
4.14.4. Integrals containing Pn (z), Jν (z), Iν (z) and Kν (z)
1
Z
1.
2.
3.
4.
-21 Z1 f -21 Z1 -
FOf
' ' % '6m _ 5 7X! ¤
¢ 3 T
G j
/ ¢ 3 / ¢
j
"' ' ' ! O'6m 5_7X! ¤ %
T G j F m G &4 - a 1 F
m
1
m ¢ 32
- a &4 F 7 ( GTj a 1
" 3 ¢ ' m % F G fI Z
Z
6.
'65 3 ¢ J ' % & L f 1 a £ / f
&4
0/ E j F m 3 ¢ G / ] f a 1 / £ f
4.14.5. Integrals containing products of Pn (z) Z
7 j j F m G '65_7 0f9 a 1 3hf - ¤
Z
7 j a 1 j a 1 F m G '65
1 1
@CB
= m
=m
£ / E j a 1 F " m 3 ¢ G
2.
Z
5.
1.
@ B
= m
f9 a 3hf - &-21 ¤ ,93,
=m
@CB @ B
& '
3.
7 j ¢ 3 £ j F ¢ 3 m G '65_7 0f a 1 3hf -; ¤
Z
1
4.
1 7
j ¢ 3 £ j F ¢ 3 m G 3
Z
5.
Z
Z
10.
@CB
f KD j F ¢ GIH
(' 'g5 7 a 5_7 #$ " " 5_7X! 1 1 N 7 #7
5 7 ( _ 5 _ 5 7 _ 5 7 a - 7 a _ - L L ! J
J
= ( 7R@CB " 5:7! ¢ f
D j F GIH
0f a 5n7 L - 5:7 L J
(' O' 5n7 a J 5n7 #$ £ 1 1 Y 0f £ N 77 = ?( 7A@ B
0f - 5n7 a 5n7 ! (
0f j
f (' 'g5 7 1 1 7 ¤ m f £ £ N ! 7 V 7 ( # , 7 5
m f &
¢ G9H
0f j D F f ( ' ]'65_7 1 1 ( m f £ £ N ¤ ! 7 # 7 ( # < 7 5
m & f
( ' ]'65_7 #7 $ 7 1 1 0 m ¢ 3 - S N & 7#7( V7< 5 ! ¤ - j
9.
7 1 7 j j F m G m 3 ¢!¤
6.
8.
=m f
Z
7.
SZ
SZ
SZ
,93 3
# #
SZ
11.
12.
Z
13.
Z
14.
Z
1
-
-
#-
Dj F
¢
5) !;
GIH
(' '65_7 1 1 ( #$ ¤ ! N 7 #7( # 7,5 &
( ' ]'65_7 #$ 7 7 1 1 j m N & 7V7 ( #7< 5 ! = m @CB
(' '65_7 1 1 ( #$ 7 ¢ !
D j F 9G H
m N & 7 #7 ( # 7,5
= m @CB 7 ¢ 3 m
j ST j 0
5_7X!A7 O'Q5_7! N
(
1 (
-S
5_7 ( ' ]'65_7 1 1 1 1 7 #$ ¤ ! & 5 1 (*' & 5n ' 1 # 7#7 #7
4.15. The Chebyshev Polynomials Tn (z) 4.15.1. Integrals containing Tn (z) and algebraic functions 1.
1 C7 ( 5
Z
4 3 ¢ ]J 1 ' L % ' ! f - &-21 f ¢ -;&-21
= m
@CB
1 C7 ( ! " ' & 3 ¢ J'6& 5_L ' 7X! % f - &-21 f ¢ -;&-!&4
a
5 m
1 = m @CB
2.
Z
3.
Z
4.
! " '& m
0f 3 -21 ¢ 3 -;&-21
3 ¢ ¢ h 3 f9
-;&-21 j F Z -21 ¢ 3 -;&-!&4 / ¢
O '6m 5_7 ¢ f O -;&-21 ,9395
¢ 3hf G
¢
/ f
@ B
=m
=m
@ B
) '
5.
6.
7.
8.
Z
-21 0fg3 -21 ¢ -;&-21 / ¢
¢ f O -;&-21 j / ¢ f
Z
-21 0fg3 -21 fg3
3 ¢ j ¢ 3 m j ¢ 3 m = '8>?7 ! m , , ) & -21 ) m ( ! " Z " m ¢ ¢ 0f 3
j ) , =m 75 m ( ! a 1
0fg3 -21 fg3
& 5_7X! 43 ¢ m 5 &L &N
J
Z
@CB
@ B
( ' ]' _ 5 7 W 1 5 & !
=m
( 7 @CB
!
0fg3 -21 a fg3
1 ( ' ]'65_7 5 #$ '65_7! m & 5 & L 43 ¢ W &
& 5nJ ! & N
& 5n !
=m !
Z
@CB
9.
=m
10.
0fg3 -21 " ¢
) & 5_7! m 5 &L &N
J
" m ( ! Z a " ¢ , 7 5 ( !
1 ) m & 5_7X! m 5 &L &N
J
Z
11.
fg3 ,
(' ' ] _ 5 7 1 5 & (
!
fg3 ,
( ' ]'65_7] 5_7 1 5 & ! (
,9 3;
=m
=m
!
!
( @CB
( 7 X@ B
( 7 @CB
# #
5) !; ,
4.15.2. Integrals containing Tn (z) and trigonometric functions 1.
Z
0f
f " 43 ¢ - 5_7 L J
Y N
2.
£ g5 7 ! f a 5_7 L 0f £ J ( ' ' a 5_7 #$ 1 ! & 1 - 5 7 a 5_7
=
( 7R@CB
0f a
f 1 " ]" '65_7! g5n ! £ 3 ¢ 0f a a & L f £ - a &L J (' J '65_ 7 5_7 a #$ & ! Y N = ( & - a & a a & &
( ' ]' #7 #$ m ! Z & ! ( ¢ f 7( m N &
F G
1 &- & a
Z m ! a F ¢ G
< 7 5
1 ( ' ]'65_7#7 & ! ( 7 ( m ! N m & 1 &- & a
'#" ' )(*'65_7! Z 0f F G a - 5 7L f - - 5_7 L J J
" #$
( £ 1;- - a 1 - 5_7 ! Y 0f £ N = ( 0f & 7 (*' - - 5_7 a - 5_7
SZ 0f
f 3 ¢ m f £ £ N 7( ( ' ]' ##7<7 5 ! m f &
Z
3.
4.
@CB
#$
¤
¤
5.
6.
,93 :
7R@CB
¤
) '
7.
8.
9.
10.
11.
12.
13.
SZ
7
f a
f 1 £ 3 ¢ m f £ £ f
SZ
SZ
0f
f m
7
SZ
SZ
SZ
f ¢ 0f
f F 0f 7
0f a 0f 1 m
m
SZ
¢ m
( ' ]'65_7 #7 #$ O'65_7 1 ! ¤ m N & & # 7 ( # 7,5
m
f £ £ f
m
f f
( ' '65_7 #7 7 £ £ & N 7 ( V7<5 ! ¤
m
f f
( ' ] ' #7 £ £ & N 7 ( V 7
f f
( ' ] ' #7 ( £ £ & N 7 ( # 7,! 5 ¤
G
m
f a F ¢ 7,5
1 0f m f £ £ m f 7 ¢ f ) 7<5 0f m f £ £ m f
(' ' V7 7 N& 7 ( V 7<5 ! ¤
,93<
( ¤
G
( ' ] '65_7#7 ( 7 ( V7<5 ! ¤ &N
,
( ;1 - #7 ( ¤
7 ( #7,5 ! &N
# #
14.
SZ
15.
16.
17.
18.
19.
20.
21.
22.
SZ
SZ
SZ
G
( 1 - 1 #7 " '' '*>?7R@CB ! m N & 7 (*' V 7 ( # 7,5 =
G
( 1- #7 7 m f £ f £ & N 7 ( #7,5 ! ¤
3 ¢ m m
f 0f
' m
V-
-
a 1
7 ¢
m
S Z "
SZ
¢
-
( ' 1 (*' #7 7 ¤ ! & N 7( V7<5
7
a m
1
= '*> @CB
¤
(' ] '65_7 1 #7 #$ ¤ ! N & & #7 ( # 7<5
( ' ]' # 7 7
3 - S & N 7 ( #7,5 ! ¤
(' 'g5 7 #7 7 ¢ 3 - S & N 7 ( #7<5 ! ¤
( ' ]' #7 ( #$ 7 ¢ ¤ ! S m 3 - & N 7( V7<5
,93 L
3 ¢ £ ¢ ¢ 3 m
SZ
f £ £ f
( ' 1 (*' 1 #7 f £ m f £ N & 7 * ( ' V7 ( #7,5 ! ' ( ' ' V7
( X 7 ! ¢ S m 3 V- & N 7( #7,5 !
S Z "
f 0f
SZ
F f £ f £ f A F 7 0 f m
f 0f m
5) !; ,
-S
) '
23.
24.
25.
26.
SZ
-
SZ
F "
7,5
SZ
-
¢
¢
a 1 F
G
¢
28.
)
SZ
SZ
-
-
-
SZ
Z
30.
Z
31.
Z
7,5
7
,
-S
( 1 - V7 ( #$ ¤ & N 7 ( V 7<5 !
R ¢
-
(' '65_7 #7 ( & N 7 ( # 7<5 ! ¤
( 1 #7 " #$ ]' " ' 1 m N & 7(* ' # 7 ( # 7,! 5 ¤ l
#$ ( 7 7 ¢ 1;- #7 ! 7 ¤ S G
A F m 3 - N & 7 ( #7<5
-
F G
#-
( ' ]' #7 ( & N 7( # 7,! 5 ¤
G
7 ¢ 3 m
¢ 3 V - S 29.
-S
7 ¢ 3 V - S m
¢ 3 # - S
27.
7 ¢ 3 m
R ¢
' ( 7 ! m
7 N 7 m & J
'#" m N l & ( ' ]' #7 ! & N 7( V7<5
( ' ]' #7 7 ( m #7,5 ! m L
7 N 7(( ' ' VV7<7 5 ! 7 m &
,93 P
( ' 7 (* O'
( ' 1 V7 #$ 1 *
#7 ( # 7,5 ! ¤
= m @ B @ B
= m
= m
@ B
# #
32.
33.
Z
Z
34.
Z
35.
Z
36.
Z
37.
Z
38.
Z
39.
40.
-
"
a 1
a 1
5) !; ,
3 ¢ O'6 5_7X! N ( ' ]'6 5_7#7 ! m 5 7 & & - & a
= m
( ' ]'65_7 1 #7 3 ¢ £ ¢ N ! m & & # 7 ( # 7<5
" ( ' ]'65_7#7 7 7 m & N 7( #7,5 !
a 1
(' ' V7 ( #$ 7 ! ¢ - m & N 7 ( #7<5
7 ¢ -
) 7,5 ,
( 1 - V7 ! ( 7 N m & 7 ( V7<5 (' ' # 7 (
7 F ¢ G - m & N 7 ( V7
" ¢ 7<5 a 1 F G
#$
#$
7 N (7 ' ( '6 5_# 7 7<#5 7 ! ( m &
= m
-
@ B
= m
@ B
= m
@ B
@CB
= m
@ B
= m
= m
F G
( ']" ' 1 #7 " #$ 1 ¢ ! = '*> ! m m N & 7( ' V7 ( #7<5 l
#$ ( V 7 7 Z 1;- 7 ! 7 G
= m - F m N & 7 ( #7,5
Z
@CB
@CB
,!5N
@CB
@ B
+ * '
Z
41.
-
']" m
( ' 1 ( ' 1 #7 N & 7(* ' #7( V7<5 !
= '*>
#$
!
@CB
m
4.15.3. Integrals containing Tn (z) and special functions Z
0 f a 1
1
1.
2.
1
Z
3.
1
Z
7! m m '65_ D V;- &-21 F G 3 # a 1 F IG H m = m
3 fI 1
-21 3hfI -21 j a 1 / E F#" m G £ f ¢ h
@CB 7R@CB
m = h
-21 3hfI -21 j a / E a 1 F#" m G £ f a 1 ¢ 3hfI 1
= hm
7R@CB
4.16. The Chebyshev Polynomials Un (z) 4.16.1. Integrals containing Un (z) and algebraic functions
j ¢ 3 £ f9
0f9 32
-21 3 ¢ P
1.
Z
2.
Z
3.
4.
0f 3 -21 3 -;&-21 F G
43 ¢ 3hf9
-;&-21 j a ¢ 3 m ,
1 ) *
=m
=m
!
m
@ B
7R@CB
&L ' 1 C7 ( ! ' & 3 ¢ J6 '
5_7X! % f - &-!& f9 ¢ -;&-21
5 m
= m @CB C7 ( ! Z1 ' 3 ¢ ' % J & '6L 5n ! f - &!- & f9 ¢ -;&-!&4
a
5 m ' &
1 = m @CB Z
,!5)
# #
5.
Z
-21 0fg3 -21 fg3
5) :1 ,
43 ¢ j ¢ 3 m , )
=m
@CB
4.16.2. Integrals containing Un (z) and trigonometric functions 1.
Z
Y
0f f
f £ £ 0f
43 ¢
( ' ]'65_7 a J
1 N & 1 - 5_7 a
" 65_7! 5 7L a 5 7L J 5 7 #$
! = ( 7A@ B _ 5 7
0f a
f 1 " '65_7X! g5: O! m 0f £ f £
' % a a &L - a &L J ( ' '65n 5_7 a #$ J
& ! Y N = ( @CB
a a a & & - & &
Z 3. f F ¢ G
£ ¢ m ! N (' 'g5 7 #7 ! ( ¤ 7 ( m &
&- & a
Z m ! a F ¢ G
4. < 7 5
1 £ ¢ 7 ( m ! N (' 'g5: #a 7 ! ( ¤ m &
&- & '#" (*'65 7!
Z 0f F G
5. a - 5_7 L
f - - 5_7 L ( - a J1 - 5_7 " J#$
£ ; 1 0 f ?')(:7A@ B ! Y = £ N
( ' _ 5 7 a 5 7 & f - - -
SZ 0f 6.
f 43 ¢ m f £ £ N (' 7 ( '65_#7,7 #5 7 ! ¤ m f &
,!5, 2.
Z
+ * '
7.
8.
SZ
SZ
0f f
£ m f f
7
f f
&43 ¢ 9.
SZ
7
10.
f 0f
11.
£ 12.
SZ
13.
SZ
7,5
7
f f
m
F m
( ' ]'65n #7 #$ £ 6 ' 5_7 1 ! ¤ £ m N & & 7 ( # 7,5 #
a 1
m f f
f 0f £
¢
f m f
f 0f
£
SZ
m
0f 0f
SZ
a 1
( ' ]'65_7 #7 7 #$ £ O '65_7 1 ! ¤ £ m N & & 7( #7< 5 #
m
F G
f f
¢
f 0f
( ' ]'65n #7 7 #$ £ 6 ' 5_7 1 ! ¤ £ m N & & 7( V7< 5 #
( ' ]'65n V7 ( #$ £ 6 ' 5_7 1 ! ¤ £ m N & & #7( V 7<5
" #$
( £ ! ' 1;- 1 V7 ! £ m N & ( ' #7 ( # 7,5
G
( ' ]'65_7 #7 £ '65_7 1 £ m N & & #7( V 7<5
¢
a 1 F G
(' ]'65n #7 1 f £ £ '65_7 N
m # 7 ( V 7<5 & f &
,!53
f f
¤
!
( #$
( #$
¤
!
¤
# #
14.
15.
SZ
f ¢ 0f £ m
SZ
f 0f
16.
17.
SZ
S Z "
a 1
f f
f f
18.
19.
20.
21.
-
S Z "
SZ
SZ
-
F
¢
£ '& £ m
¢ ¢
m 3
-S -S
" £ ¢ 7( m
a 1
¢
-
£ £
O'65_7 ¢ 3 m
,
#$ ( '65_7 ;1 - 1 #7 ! ( ¤
& #7 ( # ,7 5 m N &
7<5
)
£ 43 ¢
SZ
7
( ' V( 1 (*' 1 V7 N & ( O')(:7 V7 ( # 7,5 !
' ( ' ] 6 ' _ 5 7 # 7 ( 7! ¢ S !
m 3 - & N 7 ( #7<5
m
-
5) :1 ,
¤
¤
(' ] '65n 1 #7 #$ ¤ ! N & & #7 ( # 7<5
( ' ]'65_7 #7 7 #$ #7< 1 5 ! ¤ N V 7 ( & &
( ' ' 5* 7 7 #$ ( 1 ! ¤ N 7,( 72 5
&
&
"( ¢ 7 ( m
(' '65n 1 #7 ( #$ ¤ ! N & & #7 ( # 7,5
G
"( £ ¢ 7 ( m ,!5 5
( ' ] '65_7 1 #7 ( #$ ¤ ! N & & #7 ( # 7,5
+ * '
22.
23.
24.
25.
SZ
SZ
SZ
SZ
Z
27.
Z
28.
Z
29.
Z
Z
"
725
26.
30.
-
¢
-
-
-
-
a 1
-
a 1
¢
G
"( £ ¢ 7,( m
( ' ' 58 1 7 ( #$ ¤ ! N & & ]7,( 75
7
) <7 5 ,
( #$ 1;- 1 #7 ! ( ¤ '65_7 ¢ 3 - S N
m & & V7 ( #7<5
G
F " #$ ( ' 1- 1 #7 ! ¤ ! ¢ 3 - S N m & ( ' V7 ( #7<5
' ¢ 3 m
" "
a 1 F
a 1
( ' (')( 1 1 #7 N & ( O')(:7V7 ( V7<5 !
( ' ]'65_7V7 ( 7! ' = m m & N 7 ( #7,5 !
(' 'g5: #7 # 6 ' 5 7 ! ! 3 ¢ 5_7 N m &
&- & a
= m (' '65_7 #7 7 #$ O'65_7 1 ! = m m N & & V7 ( #7< 5
(' 'g5: 1 #7 #$ 6 ' _ 5 7 £ 3 ¢ ! m N & & V7 ( # 7<5
= m ( ' ' 58 7 7 #$ 1 ! ] ' 5n7X! N = m m & & 7( 7 5
,!5; -S
#$
¤
@ B
@CB
@ B
@CB
@ B
# #
Z
31.
Z
32.
-
Z
Z
35.
Z
36.
"
7<5
-
-
-
¢ F
34.
Z
33.
¢
a 1 F
5) :1 3
( ' '65n 1 #7 ( '65_7 N ! m & & # 7 ( # 7,5
@CB
= m ( ' ]'65_7 V7 ( #$ '65_7 1 !
G
m N & & #7( V 7<5
= m
¢
@CB
G
( ' '65n 1 #7 ( ] '65_7X! N ! m & & # 7 ( # 7,5
7 ¢ ) 7,5 ,
( 1;- 1 #7 ! ( '65_7 N m & & V7 ( # 7<5
( ' # ! 1;- 1 7 ! F G
m N & ( ' #7 ( #7,5
#$
#$
= m
#$
@ B
= m
@ B
" #$
= m (' V(')( V7 #$ ' 1 1 !
m N & ( ')(:7 #7 ( #7,5
= m
@CB
@CB
4.16.3. Integrals containing Un (z) and Kν (z) 1.
Z
0f
m m D
G9H G 3 a F F ;- &-21
1
4.16.4. Integrals containing products of Un (z) 1.
SZ
0f £ f ( ' ]'65: 1 #7 £ Y N &
& # 7 ( #7,5
,!5:
!
m 7
@ B
f £ £ '65_7! m f (' '65n #7V7 7 3 N & #7 ( #7,5 ! ¤
#$
= m
2.
3.
SZ
Z
-
, '
'65_7! 0
m (' ]'65: #7 7 #$ ( ' '65n #7V7 7 1 ! £ ¤ Y N & & 7( #7, 5 3 N & #7( #7,5 ! #
¢ 3
-S
0
( ' ]'65n V7 7 '65_7X! £ N 1 ! m & & # 7( #7, 5
-
#$
3
(' 'g5: #7 #7 7 N & #7( V7<5 !
m =
@CB
4.17. The Hermite Polynomials Hn (z) 4.17.1. Integrals containing Hn (z) and algebraic functions 1.
-21 0fg3 -21 fg3
43 ¢ O' '% ! % f 4a -21 N & &
Z
W W (' ! 1 9a 9 a a 1
#$
=m
@CB
-21 0fg3 -21 a fg3
1 W W ( ' 5 1 I5 1 #$
6 ' _ 5 X 7 ! 7 7 % £ 43 ¢ ' % f 4a F G N ! & & & 9a a 1 Ia 5 7 = m ! ( 7 @CB Z 3. 3 1 0f63
1 0fg m 3 ¢ O'6')5_(877! ! % % m £ 1 m 3 f9
, ) &-21 ) '8, >?7 C@ B m = ! '! % m Z m ¢ 4. 3 -21 fg 3
43 ' % ) 1 0fg , = m C@ B 2.
Z
,!5<
# #
5.
6.
7.
8.
Z
-21 0fg3 -21 fg3
43 ¢ ' '% ! % m , )
'65_7! % m ¢ a 0f63
43 '65_7X! % 1 ) 1 '65_7X! % m Z ¢ a fg3
43 '65_7X! % 1 1 Z
5) !< ,
m
0fg3 4a 1
Z
)
,
m ,
@CB
=m
@ B
=m
=m
@ B
0f63
W #$ ( ' ] 5n O ' ! 5 n
!
% 43 ¢ £ - -21 / 9f 4 a 4& ' % ! 5 L N
1 5
J
=m
! Z 4 a
1 0fg3 a 3
9.
1 0fg W #$ ( ' ] 5 5 L
6 ' _ 5 7 ! % 3 ¢ £ - -21 O/ f9 4a ' % ! J 5 ! N & 5
= m ! ( '65_7X! % m Z m ¢ 10. 3
43 '65_7X! % 1 ) , -21 a 1 fg =m Z ! 4 & 11. 3 3
-21 fT
fg m 3 ¢ ' '% ! % / £ m ) , =
( 7R@CB
@CB
4.17.2. Integrals containing Hn (z) and the exponential function 1.
Z
2 3hf9 O &-!&4 -
W W
,!5L
@CB
@CB
£ &-21 f9 :F 3 7 G - W W ' >?7A@ B m= ! ! 8
Z
2.
W W
2 3hf9
&-21 -
£ fI 4 F 7 G - W W a
1 = m !
Z
3.
&-21 -
W
, '
3 ¢ f -; F m G
= '*>?7
!
@ B
m
@ B
4.17.3. Integrals containing Hn (z) and trigonometric functions 1.
2.
3.
4.
5.
Z
0f a 1
'%& 3 ¢ 7 ( m ! F G m
Z
SZ
SZ
f
43V m ! F 7 G N 7( m & &
7
0f
f £ m f f
0f f £
a ¤
f a
1 f 3 ¢ £ a 1 m m
SZ
( ' #7 N &- & !
( ' #7 £ ( ! ' 7 £ m F G N 7 ( #7,! 5 ¤
f f
G
F
m f £ f £ ,!5P
(' V7 & #$ ¤ ! 1 &- & a
£ £ F G &N &
( ' 1 #7 #$ ¤ ! & V7 ( #7,5
(' 1 (*' 1 V7 ! ' ¤ m N 7 ( # 7,5 ( "
!
# #
6.
SZ
0f f
£ 7.
8.
SZ
-
S Z "
a 1
a 1 a 1 F G
(' V(')( 1 1 V7 m f £ ! ' & ¤ f £ m N 7 ( #7,5 ( "
! ' (' V7 ( ! 7 ¤ ¢ S
m F G 3 - N 7( #7,! 5
9.
10.
-
SZ
11.
Z
12.
Z
V-
G
F
' ! ¢ 3 m
13.
-
-S
a 1
#$ 1 #7 ! ¤
#7,5
( ' 1 (*' 1 #7 ( " #$ ¤ ! N
7 ( # <7 5
(' V(')( 1 1 V7 ( " #$ ¤ ! N 7 ( #7, 5
' (' #7 ( ! 7 m F G N 7( #7,! 5 = m @CB
( ' 1 #7 #$ ! m & N & & #7 ( #7,5
' (' 1 (*' 1 #7 !
F G m N
7 ( # 7,5
43 ¢ a 1 £ a 1 F G
Z
a 1 a 1 F G
'%& ! ¢ 3 V- S m
"
-
( ' F G m & N & #7 ( &
43 ¢ £ a 1 ¢ 3 # - S
SZ
5) !< 3
, ;9N
= m ( "
!
= m
@CB
#$
@CB
Z
14.
-
, '
a 1 a 1 F G
( ' ( ' ( '& ! N 7 ( # 7<5 m
#7 1 1 " ( !
#$
= m
@CB
4.17.4. Integrals containing Hn (z), erf (z) and erfc (z) 1.
2.
3.
4.
Z
Z
Z
0f
0f
Z
#-
W
( X7 ! ' '! % . ¢ j + 1 lC-;&-21
3 m , m ) &L ' J
= m ( 7X! ' '65_7! % . ¢ '65_7X! % j + 1RlC-;&-21 a 1
3 m , m ) = m 0f a 1
'65_7X! % . ¢ ¢ j + &4 lC-;&-21
3 m , = m 43 m ) L' J
'#" ' % m . m 5n ¢ 43 m 5n j + 1 lC-;&-21 ) m 5n f a 1
@CB
@CB
@CB ¤ ,
4.17.5. Integrals containing Hn (z) and Kν (z) 1.
2.
3.
']" ' m ¢ £ ' & 43 m 0f
; - &-21 ) 3 ,
Z
Z
Z
1 0 f
1 0 f a 1
']" ' 43 ¢ £ ' & -;&-!&4 3 m , m ) ' '& 3 ¢ £ ¢ '%& m , ;
= m
@CB
= m @CB m -;&-21 ) 3 , = m @CB
# #
5) !< :
4.17.6. Integrals containing products of Hn (z) 1.
2.
3.
4.
5.
6.
7.
S Ea 1 / E a 1 / fT3
( & ' 5_7X! % % '65_7X! % % ( ! 5n'6 5_7!A 5n'65: O! f SE a 0f = m Z a a 1 / fT / E 3
1 W #$ ( ' £ R a 1 2 f9 N ! =m & 1
Z 0fg 3 -21 SEa 1 / E / fT 3
43 £ SEa 5_75n! % %'6 O5_'T7 (:7X! % % f 1 0f
= m SEa Z 3 -21 S / E / fg 3
-21 0fg 43 £ SEa £ 3 ¢ £ 3 ¢ SEa f = m Z 3 -21 / E / fg 3
-21 0fg W #$ ( ' 7 £ R F G N ! =m 1
1 #7
Z 0fg 3 -21 a 1 / E / fg 3
W #$ (' 7 £ R a 1 F G F G f N ! =m 7 & 1
SZ 0f /
/
f ( ' ]'65 V 7 '& 7 £ m 1 1 ! f £ N
V 7 m F G f 1 1 & ( # 7,5
Z
, ; ,
@ B
@CB
@ B
@CB
@CB
@ B
#$
¤
8.
9.
10.
11.
12.
13.
14.
SZ
, '
f a /
1 f £ R a & F G m m SZ - / / ' ¢ 3 #- S m 7
S Z "
Z
a 1 / ¢ 3 - S
a 1 £ R a
a
1 /
(' '65 & f £ £ N V 7 f & & (
( ' ] '65
F G N V 7 1 1 &
7
#$ 1 1 # 7 ! ¤ ( # 7<5
/
( ' ] '65 V7 #$ & 1 ! ¤
m F G N & & # 7 ( # 7,5
/
( ' ]'65 #7 #$ ' 7 1 1 ! m F G N 1 1 & #7 ( #7<5
Z " / a a
1 /
1 ( ' ]6 ' 5 & 1 V7 #$ £ R a F G N #7,! 5
m # 7 ( & &
Z a 1
£ &-21 43 ¢ 3 ¢ ' % '6 5_7X! % FD -
Z
/
#$ 1 V7 ! ¤
#7,5
@ B
= m
= m
@CB
'65_7 ¤ H G -
/ £ a 1
' 5_7 ¤ £ &- '6J & 5_L ' 7 43 ¢ 3 ¢ ' % '6 5 7 ! % F D 6 H G , ;93
# #
5) L1
'#" ! % ( '! ( 1 L ' J ¤ ST F G
( * ' %
1 ( J L ( V(' W Z 1 £ _ i E S a / &N
#&-21 7 #7 ¤ S
! ( ](' #$ W Z £ ¢ £ ¢ 1!7 ¤ " * 3 3 & N
V-
S
1 ( 1 ( '
W Z 7 1 L "
a 1 £ 4 J $& % ! ¤ -
Z
15.
16.
17.
18.
W
4.18. The Laguerre Polynomials Lλn (z) 4.18.1. Integrals containing Lλ n (z) and algebraic functions 1.
7X! ' 4a -21 0fg3 -21 c fg3 4 ( P5_ ' % f 2 1 W ( ' #$ Y N 5 7 9a ! 9a a = @CB & & 1
Z 3 c c fg 3 4
c 0fg 1 IF m G c a 1 '65 P5_7X! c a 1 m ( 7A@ B , = '65 P5 & L ) J
' Z % m fg 3 4 43 ¢ '65_7X! % a 1 ) , ¤ Z m & (P5_7X! ' 5n ! 4 a 0fg 3 1 c fg 3
& ' % 5 L J
#$ ( ' ] 5n
( 7R@CB ! Y N m = !
5_7 5
' Z % m 3 3 ¢ O'65_7X! % * a 1 ) * , -21 0fg = m @CB Z
2.
3.
4.
5.
, ;!5
# . '/
4.18.2. Integrals containing Lλ n (z) and trigonometric functions 1.
SZ
m
m (' 1 #7 £ ( P5_7X! ' ¤ ! Y f £ ' % N _ 5 7 # 7 ( # < 7 5
& & f
" 65_7!A (P5_7X! ' Z 0f c
' % a 5_7
f - 5_7 J ( ' L a J 5_7 L #$ £ 1 ¤ ! Y 0f £ N
_ 5 7 a 5 7 # 5 7 f & & - ' m
SZ # ( ! 0f G ' %m c F
f ( ' ( Q(*' V7 #$ f £ 1 ¤ Y f £ N 7( #7<5 ( ! 1
SZ (' 1 ]7 #$ (5)7X! ' ¢ S 3 - ' % m & N & 5)7 7I( ! 7 5 ¤ - c
SZ G
- c F (' V( 6( ' #7 #$ ' ( ' % ! ¢ 3 - S N #7,5 1 ( ¤ m 7 (
! 1
#$ ( ' 1 #7 ( P5_7X! ' Z @CB ! #7,5 ' % = m - c
N
m _ 5 7 # 7 ( & &
( ' ' ( 6(*' 1 #7 #$ ( ! ' Z ' % m N 7( V7<5 ( - c F G
1
! = m @CB Z -21 m 5 m ( c
m (P5_7! ' ! N P5_( 7' 5 ! m 5_7 = m ( 7 @CB
1 ! ' % 5 1 L & & J
, ; ;
0f f
c
2.
3.
4.
5.
6.
7.
8.
# #
5) L1 3
4.18.3. Integrals containing Lλ n (z) and erfc (z) Z
1.
7 . 0f / E # '65 7 ! m j + 1RlC-;&-21 F ¢ 3 m G
= m
@CB
4.18.4. Integrals containing products of Lλ n (z) 1.
2.
3.
4.
5.
Z
cI0fg3
Y
Z
Sc ]0fg3 4
d ¢! ¢ f c a a 1 ESc a a a 1 0f O =
c 0fg3
5 '! % n %' %
SZ
Z
0f f
( 7R@CB
c 43]0fg3 4
& & m / ' ! %
SZ
W W #$ ( ' (P5n'65_7! 65n'65_7! ! 565n ! 5 7 c a a & 1N c a _
=
c 43
c
m
m
( 7R@CB
f f W
£ £
#$ (' P5n'65_7 1 V 7 5_7! ¤ ! Y D ' % ' H N c a 1 c 5 7 # 5 7 #7 ( V 7<5
-
c 43
c
5_7! ' ¢ (P ' % ! m 3 -
-S
W #$ (' # 5n'65_7 1 #7 ¤ ! N a 5_7# 5 7 #7 ( # ,7 5 c 1 c
c
W ' ( ' # 5n'65_7 1 #7 ( 5 7 ! ' ! ! % m N c a 1 c _ 5 7# 5 7 #7 ( # ,7 5
c 3
, ;9:
#$
= m
@CB
0 ' /
" /" £ d ¢ F 7 G F 7 G 2 c - Sc c
% ' %! S #7 $ ( V( ' 5 1 1! Y 0 d ¢ 4ST0d ¢ N = ( 7 X @ B
( * ( ' # 5 7 & 1 1
Z
6.
4.19. The Gegenbauer Polynomials Cnλ (z) 4.19.1. Integrals containing Cnλ (z) and algebraic functions 1.
2.
Z
-21 0fg3 -21 c 0fg3
43 ¢ ( &' ! % ' j + l c -21 . ¢ 3 m ¤ , ) Z (9! ' & . ¢ m ¤ ¢ c a 0fg 3
3 '65_7X! % f9 j + 1Rl c -21 ) 3 , 1
4.19.2. Integrals containing Cnλ (z) and trigonometric functions 1.
Z
0f c
f " 65_7X!A (9! ' 43 ¢ ' % a 5_7 L - 5_7 J L J
(' # 5:' a 1 Y N & 1 - 5 7
2.
Z
0f c f a " 3 ¢ ' % J Y N
f £ £ 0f 5_7 #$ ! a 5_7
1
65n !A (9! ' & ££ 0f a a & L f - a &L J ( ' 5:'65_ 7 _ 5 7 a & #$
! & & - a & a a &
=
( 7R@CB
=
( X@ B
, ; <
# #
3.
4.
5.
6.
7.
8.
9.
SZ
SZ
7
0f c
f 3 ¢ # ' (&% ! ' m m
f c f a 1 43 ¢ (&' ! % '%& m m
SZ
7
5) P1 ,
( ' P5n' V7 £ £ & N 7 ( #7,5 ! ¤
f f
f £ f £
(' # 5:'g5 7 # 7 7 ( V7<5 ! ¤ &N
0f c
0f a 1 ( ' P5n'65_7 #7 7 !' & 1 ! £ m f £ £ '6&5_ X 7 ! % N
m P 5 V 7 ( # , 7 5 & f 1
SZ 0f c F ¢ G
f ( ' P5n' V7 ( 1 ! £ m f £ £ '&!! % ' N
m P 5 # 7 ( #7<5 & f 1
SZ 7 f c a F ¢ G
< 7 5
1 f ( ' P5n'65_7 #7 ( 1 ! £ m f £ £ O '695 ! '%7& ! % N
m P 5 # 7 ( # 7<5 & f 1
SZ 7 0f ¢ c ) 7<5 ,
f ( 1;- 1 #7 ! ( £ m f £ £ ' &% ! ' N m & 5 1 # 7 ( #7,5 f
SZ 0f c F G
f ( 1;- 1 #7 " £ m f £ £ ( ' &% ! ' £ O N ( 6( ' V7 ( #! 7,5 & 7 f m
, ;9L
#$
¤
#$
#$
¤
¤
#$
¤
¤
0 ' /
10.
11.
12.
13.
SZ
-
c
S Z "
43 ¢ ( ' 9% ! ' ¢ 3 V - S m
c a 1
£ 43 ¢ ( &' ! % %' & ¢ 3 m
SZ
V-
S Z "
14.
15.
16.
SZ
SZ
-
V-
V-
c
'&! ! % ' ¢ 3 m
c a
1
¢ 3
SZ
(' # n 5 '65_7 1 V 7 #$ ¤ ! N & & #7 ( #7<5
-S
(' .65h' & 7 & N 7K( 7 5 ! ¤
-S
c ¢
¢
( ;1 - N & 7 (6 ( ' V 7
1 # 7 " #$ ¤ ! ( #7<5
' 9% ! ' ¢ 3 m
c F
7 7 #$ 1 ! ¤ 7 5
( ' # n 5 '65_7 1 V7 7 #$ &! ' & ¤ ! '65_7! % m N & P 5 1 # 7 ( V7< 5
c F G
( ' 9% ! ' £ O ¢ 3 # - S m
( ' g5h' N & g5 7E( 1
-S
-S
( #$ ¤
( ' ] 5:' N & 5 #7 ( 1
1 # 7 #7,5 !
( ' P5n' N & P5 V 7 ( 1
1 V 7 ( #7
G
O '9!! % ' ¢ 3 # - S m , ;9P
#$
¤
# #
17.
18.
SZ
SZ
19.
20.
Z
Z
22.
Z
23.
Z
24.
Z
"
7,5
Z
21.
-
c a F ¢ G
1 ( ' # : 5 'g5 7 1 #7 ( $# !' & ¢ '6& 5_ ¤ ! S 7X! % m 3 #- N & P 5 1 # 7( # ,7 5
7
¢
-
c
-
c a
1
"
-
"
-
5) P1 ,
c ) <7 5 ,
( 1;- ' &% ! ' ¢ 3 - S N m & P5 1 V7 (
( ' P5n' #7 ( 9 ! ' 43 ¢ ' % m & N 7 ( #7<5 !
1 V 7 ( #$ V7
= m @CB
43 ¢ ' (9 ! '5 & 7 N ( ' P5n '6 5_a 7 # 7 ! % m &
<& - &
c a
1
@ B
= m
( ' # 5n'65_7 1 V7 #$ £ 43 ¢ ( &' ! '%% & N ! = m @ B m & & V7 ( #7<5
( ' P5n' #7 7 #$ &! ' 1 c O'! % m N & P5 #7( V7<5 ! = m @CB
1
(' # 5n'65 7 #7 7 #$ &! ' & #7, 1 5 ! c a
6 ' _ 5 X 7 ! % N m P 5 # 7 ( &
1 1
= m @CB
c F G
( ;1 - ( ' 9% ! ' £ O N
( ' V 7 m 7 ( 6 & ,9: N
1 # 7 " ! ( #7<5
#$
= m
@ B
25.
Z
Z
26.
Z
27.
0 ' / c ¢
-
c F
-
¢
-
¢
( ' ] 5:' ' 9% ! ' N m & 5 1 #7 (
( ' P5n' '9!! % ' N m & 5 1 #7 (
G
c )
7,5
7
(
1 # 7 #7,5 !
#$
1
= m @CB #7 ( #$ ! #7,5
=
m
@CB
,
( ' &% ! ' N m & P 5
;1 1 V7
1 V 7 ! ( ( V<7 5
#$
= m
@CB
4.19.3. Integrals containing products of Cnλ (z) 1.
Z
f KD c F ¢ 9G H
( ' P5n' " " 5_7X!A &! ' 5 1 #7,5 ' % ! - 5_7 L a 5_7 L N P J
J
2.
Z
3.
SZ
a 1 # 7,5 ( #$
- #7,5 ! a
= ( 7R@CB
f D c F ¢ G9H
0f " 5_7X!A &! ' £ f ' % ! - 5_7 L a 5_7 L f £ J J (' # P 5n' a 1 V 7<5 ( #$ ( 7A@ B ! Y N =
5 # < 7 5 V < 7 5 a - 1
0f c m f £ £ m f f ( ' '65n # #7 7 #$ 9! ¤ Y D ' % ' H N # 5 V7 ( 1 #7,5 ! 1
,9:1
# #
4.
5.
6.
SZ
0f D c F
f m m
SZ
SZ
7.
Z
8.
Z
-
#-
-
f f
G9H
( ' ] '65n # #P 5 1 #7(
1 # 7 ( #$ ¤
#7
( ' ] P5n' 5 1 ] #7 (
#7 7 #$ 1 # 7<5 ! ¤
£ 9! ' £ D ' % H N
c
"( 7( m
D c F
-
¢
5) ,9N1
¢
D c F
¢
9! '
D ' % H N
G9H
"( 7( m
c
7 D ' 9% ! ' H N m
( ' ] '65n # 1 V7 ( #$ ! ¤ 5 1 ] #7 ( #7<5
$ # ( ' ]'65n # 1 #7 7 ! 5 1 ] #7 ( #7<5
= m @CB
9! ' D ' % H N
GIH
7 D ' &% ! ' H N m
( ' ] '65n # 5 1 #7 (
1 # 7 ( #7,! 5
#$
@ B
= m
4.20. The Jacobi Polynomials Pn(ρ, σ) (z) 4.20.1. Integrals containing Pn(ρ, σ) (z) and algebraic functions 1.
7X! ' -21 0fg3 -21 j + l . ¢ 0f63 5_ ' % ( ' 5 g5n'65_7
W Y f 4a -21 N & 5_7 9a 9a a 1 ! ( =
Z
,9:,
@CB
2.
&' 3 465 78
0fg3 4a 1
7 Y F !£
Z
5 7! ' . j + l ¢ 0fg3 £ - -21 ' % £ G f9 4 a 4& N ( ' 5 g5n'65_7 5n = &
5_7 5 ! (
( 7R@CB
4.20.2. Integrals containing Pn(ρ, σ) (z) and trigonometric functions 1.
2.
3.
SZ
0f j + l .
f £ fI fI fI . 0f j + l
f £ m f £ £ f
SZ
SZ
-
. j + l
Z
5.
Z
-
#-
( ' 5 g _ 5 7! ' 5n'65_7 ' %m N & 5 7V7 ( #7,5 _
1 # 7 #$ ¤
1
!
('
#$ 5 7 ! ' 5 g5:'g5 7 1 #7 ¤ ' % m N & 5_7#7( #7,5
! 1
(' 5 g5:'g5 7 #7 #$ _ 5 7 ! ' 1 . j + l ' % m N & 5 7 #7( #7,5
1
! = m @CB
¢ 3
4.
( ' 5 65n'65_7 #7 _ 5 7X! ' 1 ' %m N & _ 5 7#7( m V7<5 m 7 ¤ !
. j + l ¢
-S
(' 5 g5n'65_7 V 7 1 ' 5_% 7X! ' N m & 5_7V7 ( #7<5 (
!
#$
= m
@ B
4.20.3. Integrals containing Pn(ρ, σ) (z) and Jν (z) 1.
1 ¢ 3 ¢ f / ¢ 3 j -21 £ a a . g5n'65_7X! ' % + & & f - -21 Z
,9: 3
. + l
a
a a 1 / £ f
=
( 7R@CB
# #
5) ,9N1 5
4.20.4. Integrals containing products of Pn(ρ, σ) (z) 1.
SZ
0f j + l . 3 j + l .
f 43 ¢ £ m f £ £ 5_7X! ' g5_7X! ' ' % ! m f (' a a 1 a 5_7 5 g5n'65_7 1 V 7 Y N ¤ 5_7 g5_7 5 g5_7#7( #7<5 ! 7
SZ . . V- j + l 43 j + l
43 ¢ ¢ 3 - S 5_7 ' ! ' ! 65_7! ' % m
2.
Y SZ
5 7 5 g5n'65_7 a _ 5 g 5_7 #7 ( #7<5
1
V7 7
V-
¢
¤
#$ 1 # 7 ¤
(
m
#$
!
. G j + l F ¢ G
g 5 7X! ' 65_7! ' 43 ¢ ¢ 3 - S _ ' %! m (' a a a 5_7 5 g5n'65 7 1 Y N
_ 5 7 g _ 5 7 5
g _ 5 7 # 7 ( V7<5
! Z . . 3 ¢ 5 ' 7! ! 65_7! 4. #- j + l 3 j + l
% m (' a a a 5_7 5 g5n'65_7 #7 #$ 1 1 Y N = 5 g5 7 #7 ( #7,5 7 5_7 g 5_7
! Z . ¢ . ¢ 5. GTj + l F G
#- j + l F3 ( ' a a a 5 7
1 5 g5:'g5 7 43 ¢ 5_ ' 7X! ! g5_7X! N % m 5 g5_7#7( #7,5 5 7 g 5_7
! = 3.
. j + l ]F 3
(' a a 1 N
5_7 65_7
1 # 7
(
,9:95
m
@CB
#$
@CB
K
4.21. The Complete Elliptic Integral K (z) 4.21.1. Integrals containing K (z) and algebraic functions 1.
C7( ! " C 7 ( m ! & 1
Z
1
2.
Z
3.
Z
4.
Z
5.
6.
Z
7.
Z
9.
K
K
K
m
0fg3
f
@ B
C7 ( m ! B
( m ! 5 1 L & N 5 1 5! 1
J
= ! C7 ( m ! @ B
fg3
m
m
( m! B
( m! B
m
m ¢ £ 2 K 0f 3
f9 3 f * 3 & Z m m 3 -21 0fg3 1 K 0fg , 1) Z m 1) 3 -21 K 0f6 3
-21 0fg ,
W W #$
1 1 & f 4 a -21 N & 7 9 a I ! a a 1
& ( m ! B !
0fg3
¢ K / 3 E
-21 0fg3 -21
K f
1 ¢ -21 f
Z
8.
¢ 32
-21
K
#$ ! 1 ¢ 5 3hfI -21 N 1 5 m 1! 1L J
C 7 ( m ! = !
,9:;
m
( m! B
( m! B ( m! B
# #
10.
Z
4a 1 0fg3
K
12.
Z
Z
1
K
-21
K
#$ ( 7 1 1 5n W N& 7 5 !
m ¢ m fg3
3 * 3 m 0f
¢ F #" G
fg3
!
14.
Z
( m! B
& ( m ! B
fg3
( m m £ m m 3 , ,
) 1) Z m 3
-!&4 fT3 -21 K fg 1 ) , & ( m
1 0fg3 -21
m F" G
13.
n 5 O! 5 L J
( m ! B
fg3 £ - - &4
f9 4a &4
Y
11.
5) ,
K
15.
Z
3 -21 1 £
16.
Z
3 -!&4 -21 £
17.
Z
K
£ 3
K
" 7,5 m ( !
£ 3
K
7
!
!
B
B
7 ¤ ¢ D F G F GIH
- F G ¤
m ( ! 7,5 m ( !
m m ¢ #" * ,9: :
5
m! B
K
4.21.2. Integrals containing K (z), the exponential, hyperbolic and trigonometric functions 1.
2.
1 -21
Z
¢ 3 E
K /
1 -21
Z
0f;/ E 0f;/ E
/ ¢ 3E
5 5 1L m J _
5 7 ! N & &
3.
1 -21
Z
f / E 0f / E
K
/ ¢ 3E
! 5 1 J
F
1
4.
Z
5.
Z
6.
Z
7.
Z
8.
9.
1
-21
1
1
1
Z
-21
Z
K
/ ¢ 3
0f;/ E
0f / E
1
0f;/ E
K
/ ¢ 3
/ ¢ 3
0f;/ E
K
K
=
K
m ! 5 1 L N 5 1 ! 5 1 J
1 5 1! 5 7 5_7
W #$
=
W
N & 5 ! 5
1 1 1 L
m m G 1 F G ¤
#$
K 0f
Y
=
@CB
F m G F m G ¤ 1
/ ¢ 3E
£ 'g5 1 J ' % ! L N &
( 7 X@ B
F m G ¤
D F m G 3 F m G F m G 3 m 1 m 0f / E K / ¢ 3 F G ¤
£
@CB
m
1 F G9H ¤
&- 43f9
a 1 a 1 '6 5_7 O'6 5_7 ,9:<
!
a 1
m
C7( m ! B
# #
Z
10.
Z
11.
" " 5_7X! 0f K
a 5_7 L - 5_7 L J J
a V 7<5 #$ 1 1 1 ( 7 Y N 7 #7< 5 V7<5 ! a = ! & -
f K
0f £ " " 5_7! 0f £ f - 5 7L a 5_7 L J J 1 1 a 1 V7<5 #$ ( 7 Y N 7 #7< 5 V7<5 ! a = ! & -
f K m m 0f £ Y f £ N 7 1 ( 1 # 1 7,! 5 7 ( f &
1 1 #$ 1 ¢ 3 - S N ! K
m & 7( # 7,5
7(
5) , 3
@ B
SZ
12.
SZ
13.
-
Z
14.
-
K
1 1 1 ! N m & 7 ( # 7<5
1
Z
2.
Z
1
¢ 32
-!&4
(
K
7 (
(
K
m
!
£ 3 ,9: L
B
7 (
3 ¤
B
#$
4.21.3. Integrals containing K (z) and the logarithmic function 1.
@ B
£¤
B
3.
4.
5.
6.
1
Z
¢
f
-!&4
¢ f
K
K
1 -21
Z
1
Z
8.
Z
F f;/
N &
Ff/
¢
7
7<5 m
" 7<5 m 1
Z
C7<5
E 0/
=
m!
f 0 C7( m !
@ B
@CB
/ ¢ 3
m 5_7 , 3 m )+*
K
E
F f;/
Y 1
7<5 m
Z
"
1
/ ¢ 3
m 5_7 , = C7,5 m ! @CB m )+* m 5 1L ¢ ¢ f G K / 3 E
J _ 5 7 ! #$ ( 7 7,5 1 1 5 1 5 1 ! m B
& 5 7 5_
7 ( m
!
m f G K / ¢ 3E £ 3Pf9
<3 & 3f9 O 7<5 m B m 5 1L ¢ ¢ f G K / 3 E
J 5_7 ! 7 #7 5 1 5 1 #$
( 7 7,5 m B & 5 7 5_7 ( m ! !
m 5 7,5 m ¢ ¢ m f G K / 3 E
7<5 m B
£ 0f ¢ 1
7.
10.
m 5_7! K 5 &L 7 #7 5_7 5_7 J
( 7 Y N 5 5 ! & & & ! ( m =
Z1 ¢ 3 f K / ¢ 3E £ ¢ 3hfI K / f< 3 !- &4 h 1 -21
Z
Y
9.
N & F f;/
FOf;/
¢ f G K / ¢ 3 E
F f ¢ f G
,9: P
7<5 m
B
# #
11.
1 -21
Z
,7 5 m 7( m
1 # 7 5 N & _ & 5 7
¢ 3E
K /
1
Z
13.
Z
14.
Z
7<5 m 7 ( m
1 7
1 -21
1
15.
Z
16.
Z
-21
Z
17.
18.
19.
Z
Z
7,5 7 (
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K 0f
m 5 m (
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m 5 m (
&
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m 5 m (
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f
( 7
7( m
7 ( m
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E fI 0
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B
¤
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m 5_7 m
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¢ 3hf 3 ¢ G
7 ( m B m #$ m J L N 1 1 !
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Y 12.
¢ K / 3 E
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¢ h 3 f 3 ¢H 7 ( m B
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_ 5 7 D f FOf
20.
1
Z
21.
m 5 m (
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1 -21 F f /
K
Z
1
Z
m D 3
, 2D ¢ 3
1
Z
25.
¢ f G
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= & C7 ( m ! H
E 0fI
7<5 m
m 5_7X! 5 &L J
( 7 7,5 m !
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¢ f GIH
B
7 ( <7 5 m £ 3 ,
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¢ 3hf f FOf-
32
32
f ¢ ¢H
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¢ f G K / ¢ 3E
22.
7 #7#7 5 7 5_7 Y N ] 5 5 (
m & & & !
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24.
5_7 G
KF
f ¢G 3
B
7,5 m
B
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4.21.4. Integrals containing K (z) and inverse trigonometric functions 1.
1
Z
3 E
f;/ E K / ¢
m £ f <3 f 0
&
2.
1
Z
/ E
K
/ ¢ 3E F G ¤ , <
7 ( m
B
# #
1
3.
Z
4.
Z
"
1
/ ¢ 3E
7 ( m
7 7 7 E F G 3 F G ¤
f;/ E K / ¢ 3 E
m 5 1L J 5 7 !
7#7 5 1 5 1 #$ = N & _
& 5 7 5 7 ! m
" Z1 ¢ 7 ( m f;/ E K / 3E
Y
5.
K 0/
5) , 5
( 7
f
!
7 ( m
7 ( m 7 7,5 7 ( m ¢ 3 m 7 ( m f;/ E K / 3E
7 ( m 1 1 9a 1 m 9a 1 L 2- 1 K f
7 5_! 7 J L N & 7 5_ 7 ( J
m
6.
7.
1
Z
1
Z
8.
9.
Z
@ B
B
B
B
1
Z
K
1
1
10.
Z
11.
Z
12.
Z
1
1
£¤
K 0f K
7 £ f9
<3 f9
0
&
K
D 3
!
K 0f
7 ( m
B
7 & - F G ¤
7 £¤
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¢ 3hf 3 7,5 7 ( m 3 ¢
, 7 ( m
B
13.
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14.
15.
"
1
m
_ 5 7 m
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K
9 a 1 L J L J
( 7 ,7 5 m !
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1 1 Ia 1 ( m ! Y N 7 5_7 5_ 7 &
Z1 m m 5_7 K F m 5 7 G
Z
7 £ 43f9
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3 E
0f / E K / ¢ #7 5 5 5 1 1! m J 1 L N 1 5_7X! &
& 5_ 7 5 7
(
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1
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K
/ ¢ 3E m D £
¢ f
7
EF
!
1
Z
18.
Z
19.
20.
1
/ E
0f / E
1
Z
K
B
7,5 m
B
B
m
7<5 m
7 7 / ¢ 3 E D F G - F G 3 £ &4 H ¤
K
/ ¢ 3E m D0 ¢ f9
1
EF
m 5_7! 5 &L 7 #7V7 5 7 5_7 m J
( 7 Y N ! & 5 & 5 &
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7 ( 7( m 7,5 7 ( m £ , 3 ) ,
)
1 -21
Z
7<5 m G 3 H
17.
m
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Z
& 3Pf9
16.
B
f / E K / ¢ 3 E
, <93
m
7<5 m G 3 H
7<5 m
7( m
7 ( m
B
!
B
B
# #
1
Z
21.
-!&4
f;/ E
K
/ ¢ 3 F
5) , ;
¢ 3hf f
1
22.
Z
23.
Z
1
-21
-!&4
/
0/ E
K
K
3 ¢ £ £ ¤
/ ¢ 3
/ ¢ 3
:3 £ ¤
fT3 ¢ G 7 ( m B
4.21.5. Integrals containing K (z) and Li2 (z) 1.
2.
m 5 7! ¢ K / 3 E 5 & L 7#7 #7 5_7 5_7 J m Y N 5 & 5 & ! ( 7 !
Z1 -!&4 0 f K / ¢ 3E £ £ 0fg3 ¢ K 0 / f 1 -21 0 f
Z
1
Z
-!&4 4 3f Y £ E 1
4.
Z
5.
Z
-!&4 1
K
K
-!&4 4 3
m _ 5 7 , m ) *
K
B
f <3 7 ( m B
/ f ¢
/ ¢ 3E
m 5 7 , 3 m ) *
7( m
E /
3.
3 £
7,5 m
B
/ ¢ 3E £ P3 ¤ / ¢ 3E £ / 2 F G 3 £ ¤
K
4.21.6. Integrals containing K (z), shi (z) and Si (z) 1.
1 -21
Z
0f;/ E
K
f;/ E
/ ¢ 3E
5 1L m J _
5 7X ! & N
1 5 1 5 1!
& & 5 7
5 7 ,
W #$
=
( 7 X@ B
2.
1
Z
¢ f;/ E K / 3 E
K
D f F m G 3 £ F m G F m G hf F m GIH ¤ 1 1
4.21.7. Integrals containing K (z) and erf (z) 1.
1 -21 0 f;/ E
Z
/ ¢ 3 E
5 5 m 5 1L ( m 1 1 1 ! J 5_7X! N 5_7 5 7 & & &
¢ 3
0f;/ E K / K
#$
=
2.
W 1 -21
Z
m 5 1L
7 5 1 5 1 m ! J 5_7! & N & & 5_7 5 7
#$
=
( 7 X@ B
( 7 X@ B
4.21.8. Integrals containing K (z), S(z) and C(z) 1.
1 -21 f;/ E
Z
/ ¢ 3
5 & 7 m L J 5 N * & J L K
2.
1 -21 0f / E
Z
/ ¢ 3
5 " m J 1 L N
5 & & J L K
W
& 5 & 5 & ! ( & 5 5
#$
=
( @ B
=
( 7 @ B
1 5 1 5 1! ( 1 5 & 5 &
W #$
4.21.9. Integrals containing K (z) and γ(ν, z) 1.
1 -21 f K / m 5 5 5 J
Z
!
¢ 3
5 5 ( m N & 5_7 5 5 1 ! 5 5 1 & 1L
, < ;
= 5 !
@CB
+
2.
1 -21
Z
f K / m 5 ! 5 5 1L & J
5) , N
¢ 3
7 5 5 m N & 5_7 5 5 1 5 ! 5 1
= 5 !
@CB
4.21.10. Integrals containing K (z), Jν (z) and Iν (z) 1.
1 -21
0 f;/ E f;/ E
Z
K
Y 0 f;/ E f;/ E
K
/
1 f;/ E
K
/
1 0 f / E
K
1
2.
Z
3.
Z
4.
Z
5.
Z
1
1
1 -21
f
1
Z
7.
Z
8.
Z
0 f;/ E
f;/ E
N & 5
5
a
/ ¢ 3 m H fI <3hf / ¢ 3 m f L2- 1 fI 3
K
K
m 5 L !& 5 7!
5 J a 1 L _ J W #$ 5 ! = 5 !
1 5 a 1 5_7
L fI ¤ H 0fI
/ ¢ 3
/ ¢ 3 m
Y 6.
/ ¢ 3
K
7 / ¢ 3 m
7 / ¢ 3E m £
fI 0 ¤
L
5 ! m 5 5 G F _ 5 7 ! 1L 5 J 5 m #$ ! = 5 ! 5 1 5 5 1
£ ¤ fI £ fI
7 ¢ £ If f £ If 3 ¢ ¤ /
f;/ E K 3E
m 1 V1 0f;/ E
¤ -21 fI 0
1 1
5 1 5 7 5 &N & _
H
@ B
, <9:
fT3hf
£ fI 0 ¤
@ B
9.
1
Z
f;/ E 1
Z
10.
V f;/ E
1 7
Z
11.
1 7
Z
12.
/ ¢ 3 E
3 C 7 3 7 3 7 m m V1 0f;/ E
1
1
Z
14.
/ ¢ 3E
C3
£ fI <3 f
7 / ¢ 3E 7 m 3
K
1 0f;/ E V1 f / E
K
K
7
£ fI
m
7 / ¢ 3E 7 m '¢ ¡ f
£ fI £ £ ¢ £ ¤ f9 3 If
£ fI £ £ ¢ f9 f
7 m
K
£ fI
7 7 3 m 7 m
K
#
7 / ¢ 3E 7 m
K
1 0 f / E
Z
13.
K
£ fI
£ fI
£ fI 3
£ fI 3 ¢ ¢
#f9
7 / ¢ 3E 7 m ¢ ¡ f
£ fI 3 ¢ ¢
1.
1 -21
H L
f;/ E 0f;/ E
1 H f / E
K
1 L 0 f / E
K
Z
3.
Z
£ fI ¤
£ fI ¤
£ If ¤
£ fI £
3 #f fI
Y 2.
£ fI
4.21.11. Integrals containing K (z), Hν (z) and Lν (z) Z
£ fI 3
£ fI
£ fI ¤
&N
K
/ ¢ 3
/ ¢ 3E
7 5 & 5
a 1
& 5
¢ m 3
5 a 1L
/ ¢ 3 m 0fI 3 ¢ ¤ , < <
'&
L 5J 5_7 L J 5 & L J J
W 5 a 1 #$ ! = 5 ! 7 5_7 5 5_
0fI ¤
¤
( 7A@ B
+
1
4.
Z
5.
Z
1
H
0f;/ E
K
L
0f;/ E
K
/ ¢ 3 m f 3
5) , !,
¢ ¤ 9 3hf fI ]f 1 0fI
/ ¢ 3 m & ¢ f9 O 0fI 3 #f 1 0fI 3hf9 3 ¤
4.21.12. Integrals containing K (z) and Lλ n (z) 1.
1 -21 c 0f
Z
K
/ ¢ 3
!& ( 5 7! ' N P5_(7 ' 5 ! m 5
1 1 ' % 5 1 L & & J
=
@ B
4.21.13. Integrals containing products of K (z) 1.
1
Z
7 (
7
K
" 3 7 F 7 G F 7 G N £
1
2.
Z
3.
Z
7 (
1 -21
K f
K
7 N
/ E
K
/ ¢ 3
#$ 1 1 1 1 V7 & & & & ! 7 V7 #$ ¤ & & & & F G N 7 !
#$ 1 1 1 1 1 1 ¤
1 # 7V 7# 7 #7 V7 7 ! ! 1 1
5 1 L N & 7 5 J
5.
6.
/ ¢ 3 m 0 fI 3 43fI 0 Z1 ¤ ¢ K / E K / 3 E
1
Z
1
Z
K 0f;/
E
K
7 ( ¢ 3hf9
7<5 m m m ¢ f9
fI <3 43fI , <9L
¢ 3E K 0f;/ E K /
4.
!
m #$ ! 1 5 1 7 ( m B
7 ( m
7 ( m
B
B
1
7.
Z
8.
Z
9.
Z
K 0/
1
E
K 0/
K
E
K
"
1
7<5 m
/ ¢ 3
7 ¤
/ ¢ 3
O7 7 ¤
KF
m 7,5 m G
K
#
( #$ ! m 1 1 5 N & 7 5 ! 5 1 1 1 J L
7 Z1 ¢ 3 7<5 K )+* 7,5 , K /
1
11.
12.
Z
7<5
1
K
K
"
)+*
7<5
Z
13.
K
) * K
,
K
7,5 ,
K
1 1 & &
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7,5 m
B
¤
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¤
G
" 3 7X F 7 G
£ 4
7
F G N
7,5
G
¤
/ ¢ 3
Z
/ ¢ 3
10.
E
1 1 V 7 & & 7
!
#$
3
N
& & # 7 #$ ¤ 7 11 11 !
4.22. The Complete Elliptic Integral E (z) 4.22.1. Integrals containing E (z) and algebraic functions 1.
C7( ! " C 7 ( m ! & E
! ¢ 3hfI -21 N 5 7 ` ! m 5_7X! 5 1 L
1^ J
= ! C7 ( m ! @ B 1
Z
, <9P
# #
2.
3.
1
Z
7 (
Z
E 0f
£ ¢ ¢ m D f2 9f 3hf
-21 0fg3 -21
5) , ,
E
Vf9 3 ¢
W W ( 1 1 & f 4 a 2- 1 N & 7 9a I a ! a 1
m
!
4.
5.
6.
8.
#$
( m! B
m m ¢ E 0fg3
m * 3 & ( ! m m B ! Z m m m ¢ E 0f63
m * 3 & ( ! m m B ! Z
Z
Z
m
2
E
7.
C7 ( m ! B
fH
fg3
0f63
& f 3 ¢
1 0fg3 1
E
m m £ ¢ f ] ]f * 3 & ( ! m m B ! m m 0fg3
3hf9
1 ) , ( ! m m m m B 3 , , !
m £ f9
)
& )
m m 7 P3hf 1 -21 0fg3 1 E 0f63
, ) ( ! m m m m m f
, 3 , ! & ) )
Z
,9L N
B
#
E
m 3 f9
1 ) 9. h -21 0fg3 -21 E fg3
, m m m ( ! m m f
, 3 , ! & ) ) " m ( ! C" Z 10. 3
7 ( m ( ! E 0fg W W #$ 1 & &f 4a -21 N ! Ia a
7 9 a & 1
( m !
m Z
7 7 ( m (
!
m fg3 ] ( m m ! Z m 3 ] ( m 12. 7 ( m ( ! E fg m ! m ( ! Z 13. 7 ( m ( ! E fg 3
( m ! £ f9
m <3hf9 m 3 m
1 m ! m ( ! Z " 14. 3
7 ( m ( ! E fg ( m £ £ 3 f9
m m m , m
) , 1) m ! m ( ! " Z " 3
15. 7 ( m ( ! E fg # ( ! £ £ 3_f m m m , m
) , 1) m ! 11.
Z
!
E
B
B
,9L1
& ( m ! B
& ( m ! B
m
m & & ( m ! B
3
m
3
m
m
,
& ) ( m! B
m
,
& ( m ) ! B
# #
16.
Z
4a 1 0fg3
( 1 & N 7
Y 17.
Z
1
Z
-21
E
E
1
5
7 m £f m *
18.
19.
Z
20.
Z
21.
Z
E
5n O! fg3 £ - - &4
f9 4a &4 5 L J
# $ 5n ( 7 ( !
m m B W ! ! !
¢ F #" m G ] f 3 ¢ ¢ 3 m * & ( ! m m B !
fg3
5) , ,
fg3
+ m 7 * m
m ¢ F # " G * 3 m
( m! B
1 0fg3 -21 E fg3
m ¢ 3 m m 3 ¢ 3 £ f
m 3 m m 7X ) , 1 ) , ,
) & ) , & ( m m ! B ! fg3
m m m m m , ) , 3 & ) , & ( ! m m B ! m & 5n ! 3
! E fg & 5 L J
#$ 1 & 5n W ( 7 ( m! B
m
7 ] 5 ! ! !
m m 5n ! ¢ m -21
3
3 , ! E fg ) m 3 F " G m ! ( m ! B
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& m (
7 (
Z
7 (
!
m (
Y 22.
m (
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Z
23.
Z
24.
Z
25.
Z
26.
1
Z
27.
" 7 ( m ( !
#
E
m fg3 * ( m & ( m m ! m ( ! " ¢ m -21 m 3
7 ( m ( ! E fg ) 3 , Y ¢ f
m , 3hf m , 3 m m ,
)
) & ) & ( m m ! " m ( ! " 3
7 ( m ( ! E fg ¢ 3 m -21 9 ¢ 3hf
m ) , , 1) ( m m m m m f , 3 & , ! ) ) " m ( ! 5 7<5 E
m ( ! 7,5 m ( ! * & 5 m m ! ! -21 ¢ f E / ¢ 3 5 7! 5 1 L ( 5_7 m J
C7 Y N ! ! & 5 1 5 & =
E
!
B
!
B
!
m m !
B
B
( m ! @CB
4.22.2. Integrals containing E (z), the exponential, hyperbolic and trigonometric functions 1.
2.
1 -21
Z
1 -21
Z
E
/ ¢ 3
0 f / E E / ¢ 0f / E m 5_7! 5 5_7X! J 5
5_7 ! 5_7! 5 1 L 5 & L N & 5 1 5!
J J
m
3
5 1 5 &
7! N &
! & 5_7 5n
1L
,9L 3
W #$
=
&
=
@CB
( 7 X@ B
# #
3.
1 -21
Z
f;/ E 0f;/ E
E
/ ¢ 3
! 5_7X! 5 1 L J
¢ m D F 0 f / E E / 3
N
1
Z
5.
Z
6.
Z
7.
Z
8.
Z
9.
Z
10.
1 7
1 7
1
1 7
0f / E
1 7
Z
f;/ E
0f;/ E
E
f;/ E
E
/ ¢ 3
f;/ E
E
G
m D F m G
/ ¢ 3
Z
Y
0f E
( 1 1 N & 7 # 7,5
m
1 F GIH ¤
D F m G 3
m
1 F G9H ¤
3 £ - &-!& 43f9
J
')( 1 L '65 1 L
' % ! J
')( 1 '65 1 '65 1 N& 'g5 7 O'65 7 m = 7 ( m ! " " 5 7!
- 5_7 L a 5 7L J J
a 1 V7<5 #$
( 7 7 ( ! =
# < 7 5 a ! -
Y
11.
@ B
m
1 F G9H ¤
E f
D # F m G # F m GIH ¤ 1
D F m G F m G 3 m 1
/ ¢ 3
m
=
m D # F m G 3 # F m GIH ¤ 1
/ ¢ 3
/ ¢ 3
E
£
E
W 5 7 #$
_ ! & 1 5 1 5 &
m m 1 F G 3 V1 F G9H ¤
4.
5) , , ,
@CB
,9L95
@ B
"
Z
12.
7 (
Z
" " - 5_7 L J
0f E
1 & a 1 5 N & 7 5_7 a -
f E
0f 5 7! 0f a 5_7 L f J
0f 7,(
f " " 5:7! - 5n7 L a 5:7 L J
J
Z
14.
E
15.
Z
E 0f
16.
17.
18.
SZ
SZ
E
" " 5 7! a 5 - 5_7 L J
J
= ( 7 7 (
7 ! 5_7 #$
!
7L
f £ £ 0f
f 0f
E
3 £ -;&- f
E-Q1 & N 5_7
m ( Y N 7 &
,9L;
#$ 1 & a 1 5n7 ! N
& 7 - 5n7 a 5:7
= ( 7 ! 7( @ B ' E-Q1 a 1 J L 5_J 7 L L a 1 J a 1 7 ( m C@ B '6 5_7 m = !
f £ £ f 1 1 1 7 ( ( #7, 5 ! =
( 1 1 1 #$ ¢ S m 3 - & N 7( V7< 5 ! - E
7 ( = " 1 & ¢ 1 S m 3 - & N 7( # 7,! 5 7 ( E
7 ( =
@ B
( a 5_7 #$ £ 1 1 1 £ N & 7 5 7 a 5 ! 7 -
= ( 7 ! 7( @ B
Y
SZ
Y 13.
#
m
@CB
@CB
@CB
#$
# #
Z
19.
E
-
( 1 1 1! N m & 7 ( # 7< 5
5) , , 3 #$
m
!
7 (
B
4.22.3. Integrals containing E (z) and the logarithmic function
1
1.
Z
2.
Z
¢ 32
-!&4
1 -21
¢ f
E
3 ¤
/ ¢ 3
m 5_7X! 5_7X! N 7 #7 5_5 7 5n5
& 5 ! 5 & L &
J
=
E
1
3.
Z
4.
Z
5.
Z
6.
7.
1
-!&4
¢ f
E
-!&4
¢ 3hf
E
/ ¢ 3 £ 0f ¢ 1
/ ¢ 3 £
E fI <3
E
!
( m
( 7
!
& C7<5 m !
@ B
m 5 _ 7 , 3 ,
m )* = C7,5 m ! @CB = & C7 ( m !
,
@CB
m 5_7X! 5 1 L ¢ f G E / ¢
F f / 3 E 5 7! J 5 7!
1 5 1 5 & #$ 1
( 7 7,5 m B Y N m
_ 5 7 n 5 ( !
& & !
Z1 ¢ ¢
F f;/ f G E / 3E
9 f9 ¢ 43f9 O 3 9 ¢ 3 £ f9
43f9
<3 ]f9 43f9
0 m
1 & 7<5 m B 1 -21
1
Z
7
7<5 m
F f;/
£ f9 3 £ m )
¢ f G
E
¢ f £ f9
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/ ¢ 3 E
7,5 7<5 m ,
7<5 m
B
8.
9.
"
1
Z
7<5 m
1 -21
Z
Y 10.
7<5 m 7 ( m
1
11.
Z
12.
Z
1 7
7<5 7 (
7,5 m 7 ( m
7<5 m
7( m
7 ( m
1 # 7 5 N & _ & 5 7
¢ 3E
E /
/ ¢ 3
E
1 5 &
5n m
!
K 0fI
#$
( 7
9 ¢ 3 £ f9 O E fI 0
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1 7
Z
14.
Z
15.
16.
1 -21
1
Z
1
Z
7
7,5 7 (
B
E
B
B
3 ¤
/ ¢ 3 m 0f9 3 ¢
K fI
E fI 0
13.
E
7<5 m 7 ( m 9f 9 f9 3 ¢ m 1
Z
E
¢ f G E / ¢ 3E
F ¢ f 3 ¢ G
m m 5 7! 5 1 L ¢ 3E
E / # 5_7X! J 5_7X!
F f;/
#
7 ( m
B
/ ¢ 3 £ ¤
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( L m #$ 1 1 ! E f
IJ a 1 L N & 7 9 a 1 5 7 J 7 ( m @ B = !
,7 5 7(
7( 7 ( E f
D f f- f9 £ ¢ 3hf 3 £ H 7 (
m 7<5 7 ( F ¢ 3 ¢ h 3 f G
7 ( m ( 7 ( E 0f
m 7 ( ,9L<
m m
B
B
# #
7<5 7 ( 7<5 m 7 ( 7 ( 1
17.
Z
18.
Z
-21
m 5 m (
E
m
5) , , 3
F m
m
¢ f 3 ¢G
7<5 m B
7<5 m G
EF
m
! 5_7
( L 1 1
9a J 1 L N & 7 I a 1 J
m
19.
20.
Z
Z
m 5 m (
7X D f
&
E
m 5 m (
7 ( m
7 ( m
B
0f O 0f9 £ ¢ h 3 f 3 £H 7( m m !
B
!
7
0f O Vf 3 ¢ 9f # ¢ h 3 f 3 H 7( m m B !
"
Z
#$
m 5 m (
Y D ¢ ]f ] #f9
21.
E
E
L m #$ 1 & !
Ia J 1 L N & 7 Ia 1 5_7 J
m !
22.
Z
m 5 m ( 7 ( 7
E
m
,9L L
7 ( m
B
7( m
B
1 0f
m
!
23.
Z
1
24.
Z
25.
Z
Z
26.
m 5 m (
7 (
7 (
7
7,5 7(
E
¢ m 5 m (
7 D f FOf ' &
F ¢ 3 m
E
¢ 3hf G
7( m !
7 7 7 F G ¤
E
1 -21 F f;/
Z
27.
¢ f G E /
m 5_7! 5_7! N 5 ! 5 & L J
¢ 3E
7V7#7 & 5
=m !
1
Z
28.
-!&4 F f;/
)
_ 5 7 & 5
(
¢ f G E / ¢ 3
¢ f 3 7,5 ,7 5 m 3 ¢
,
B
¢ D # f ] Vf9 3
B
FOf ' f ¢ G 7<5 ¢ ¢ m m #f f V f 3 H ! 7
5 7 G
_
EF
f ¢ G 0f 3 £ f ¢ £ H , 7 5 m m ! m 5 m ( E F 5_7 G
¢
#
5
n ! ( m 7 ,7 5 m !
B
@CB
=m
!
7,5 m
@CB
4.22.4. Integrals containing E (z) and inverse trigonometric functions 1.
1 -21
Z
Y
f;/ E E / ¢ 3 E
5 1L 5 &L J # _ 5 7X! J 5_7X!
1 1 5 1 5 & N & 5_7 5n m & !
m
,9L P
#$
( 7 !
7( m
B
# #
2.
3.
1
Z
1
Z
f;/ E E / ¢ 3 E
&0f9 3 ¢ f9
9 £ f9 ¢ f9
<3 ] f9 f9
0 m
1 & 7 ( m
4.
5.
Z
6.
Z
1 7
7 7 7 f;/ E E / ¢ 3E 7 F G F G
1 7
Z
7 ( m
8.
1
Z
7
7 ( m
f;/ E E / ¢ 3 E
B
7 # 7 5 N & 5_7 &
1 5 &
5n m
!
f;/ E E / ¢ 3E
5 1L 5 &L J ] _ 5 7! J 5_7!
m
#$
( 7
m ,
F¢ 3
( 1 9 a 1 L 9. E f
J L N & 7 J
Z1 ¢ £ £ ¤ 7 10. E
1 -21
B
¤
f
,7 5 7 ( £ 3hf9 3 £ ¢ h 3 f 3 £ f9 m ) " Z1 ¢ f / E E / 3 E m 7 ( m Z
7 ( m
7 7 7 / E E / ¢ 3E 7 F G 3 F G ¤
"
1
m
0 f;/ E E / ¢ 3 E
Y 7.
5) , , 5
,9P N
7( m
7 ( m
!
¢ 3hf G
7 ( m 1 9a 1 ! m
5_ 7 5_7
7 ( m !
B
B
B
B
11.
Z
12.
Z
13.
Z
1
1
2
E
14.
( 1 N & 7 5
"
1
Y Z
1 ¢ -21 f
£ ¤
m
E F 7<5 m G
1 9a 1 7 5_7 ( m
!
7 ( m E 0f
1 & 9a 1 9 a 1 L J L N & 7 5_ 7 5_ 7 m J
! 7 Z1 7 ( m E 0f m f 1
7 (
1
7 (
1
7
Z
17.
Z
18.
Z
E
E
7<5 m
EF
7 ( m
7 ( m
!
19.
1 -21
F f m
m
7<5 m G
0 f;/ E E / ¢ 3 E
Y
m
B
B
¢ f G
7<5 m
B
5 1L 5 &L # J 5_7X! J 5 7!
1 # 7 5 1 5 N & 5_7 n 5 ( & !
7 3 £ £ ¤
Z
B
7 7 D F G 3 ¢ F G9H ¤
16.
9 a 1 L J L J
7,5 m !
15.
E
7 7 7 F G F G ¤
E
#
,9P1
&
m
#$
!
7,5 m
B
# #
1
Z
0f;/ E E / ¢ 3 E
m m 7 5_ 7X! 9 £ 9f ¢ E F m _ 5 7 G 3 m
20.
1
21.
Z
22.
Z
1 7
/
1 -21
Z
23.
E
/ ¢ 3
f;/ E E / m 5n7X! 5 ! 5 J
1
Z
f;/ E
E
, ¢ 3 )
1
Z
25.
-!&4
-!&4
/
E
¢ 1
m 5 7 GIH
7<5 m
B
¢ 3 E
7 77 5n7 5* 5n7X! N O 5 & 5 m &L & !
?( 7 ,
7 ( m !
B
/ ¢ 3
¢ 3hf 7<5 7( m
,
/ ¢ 3 , ¢ 3
m K F m 5 7 G 3 m 5_7 7<5 m B m
3 7X7 " D F 7 G L¢ £ F GIH ¤
0f;/ E E / ¢ 3E m 0f9 Y D K F m 5_7 G 3 E F m m
24.
5) , , ;
7( m
B
£ ¤
4.22.5. Integrals containing E (z) and Li2 (z) 1.
1 3 E
-21 0 f E / ¢ m 5_7X! 5_7X! N 7 # 7 #7 5 5 7 5
& 5 ! 5 & L &
J
=
Z
,9P,
5
n ! m ( 7 C7 ( m ! !
@ B
#
E
4.22.6. Integrals containing E (z), shi (z) and Si (z) 1.
1 -21
Z
0f;/ E
E
f;/ E
/ ¢ 3E
m 5_7X! 5 1 L 5 7! J 5 7! & N
2.
1
Z
0 f / E E /
W
1 5 1 5 &!
& & 5_7 5n
#$
=
( 7 X@ B
¢ 3E
D f9 F m G 3 £ f F m G F m G ¢ f9
F m GIH ¤ m 1 1
4.22.7. Integrals containing E (z) and erf (z) 1.
1 -21 0 f;/ E E / ¢ 3E
m 5_7! 5 1 L 1 ] 5 7! J 5 7! & N & &
W Z1 -21 0f;/ E E / ¢ 3E
m 5_7! 5 1 L ] 5_7! J 5_7! & N & &
Z
2.
5 1 5 & 5_7 n 5 ( m ! #$
7 5 1 5 & 5_7 5n m ! #$
=
( 7 @CB
=
( 7 X@ B
4.22.8. Integrals containing E (z), S(z) and C(z) 1.
1 3 E
-21 f;/ E E / ¢ 5 ! 5 & L 7 m J 5 O ! 5
* J L Z1 -21 0f;/ E E / ¢ 3E
5_7X! 5 1 L m " 5 ! J 5 & L J Z
2.
&N
W
& 5 & 5 ! ( & 5 5
=
1 5 1 5 ! ( & N 1 5 & 5 ,9P 3
#$
( @CB
W
#$
=
( 7 @CB
# #
5) , , P
4.22.9. Integrals containing E (z) and γ(ν, z) 1.
1 3 E
-21 f E / ¢ 5 5 5_7 ( m m 5 ! 5 !
5 5 1 L & N & 5_7 5 5 1 5 ! 5 & 5 5 7! J
= 5 !
Z
2.
1 -21
Z
f E / ¢ 3 E
7 5 5 _ 5 7 m m 5 ! 5 !
5 5 1 L & N & 5_7 5 5 1 5 ! 5 & 5 5 7!
J
= 5 !
@CB
@CB
4.22.10. Integrals containing E (z), Jν (z) and Iν (z) 1.
1 -21
0 f;/ f;/ m ! & 5 5 Z
1 0f;/ E
2.
Z
3.
Z
4.
Z
5.
Z
1 7
1 f;/ E
1 f;/ E
1 7
E
1 0 f / E
/ ¢ 3 m f
E
E E / ¢ 3
E W #$ 5
5 ! 5 L 5 5_7 ! 7! 5 J 5_ 7 L 5_7! N & 5 a 1 5 a & 5 7 J
= 5 O! @CB
E
/ ¢ 3 m
H
0fI 3
H
/ ¢ 3 m
L
,9P95
¤ 1 fI 0
¤ 1 0fI
/ ¢ 3 m f L 0fI 3 E
H
¤ 1 0fI
L
¤ 1 0fI
6.
1 -21
Z
/ ¢ 3
5 ! 5n 5 7! 5 J 5 1 5 Y N 5_ 7 5 & &
f
1 -21
Z
f;/ E 0f;/ E E / f;/ E f;/ E 565 ! 5 5 g5 5_7X! J 5 a a a J 5_7 Y N 65_ 7 1 5_7 6 5 5
1
9.
Z
1
E
1 f;/ E
E
1 7
10.
Z
11.
Z
12.
Z
13.
f / E
1
0f;/ E 1 1 7
Z
/ ¢ 3
E
E
V1 0f / E
E
7
m f
&
= 5 !
@ B
#$ m ! a a &
= 565 !
5_7
@CB
£ fI #f9 £ If 0 ¤
£ fI <3
m (& 5_X7 !
£ fI Vf £ If ¤
7 / ¢ 3 m £ If £ f9 3 ¢ ¤
/ ¢ 3
E
#$
m
£ £ m If 3 f
/ ¢ 3
m F G
7
/ ¢ 3
1 0f;/ E
1 V f / E
5 ! 5 1 L 5_7!
5 5 7! 5 1 5 5
a L " ]" " 65_7!
a a 1 L
5 a 5 a 7 5 a a 1
5
Z
E
¢ 3
8.
E
7.
7 £ m f
7
m f
,9P;
£ fI 3
7 / ¢ 3 m
£ fI 3
£ fI #f9
£ fI 3 #f9
£ fI 3 £ f9 3 ¢ ¤
£ fI ¤
£ fI ¤
+
5) , ,
4.22.11. Integrals containing E (z), Hν (z) and Lν (z) 1.
1 -21
Z
H L
f / E 0f / E
Y
1 H f;/ E
E
1 L 0 f;/ E
E
2.
Z
3.
Z
4.
Z
5.
1 7
H
0f;/ E
1 7 L 0 f;/ E
Z
&
/ ¢
/ ¢ 3 E
5 5_7X! 5 5n O! 5 7 5 J a N 5
& &
3 m ¢ E
5 a 1 m a 1 J 7 5L G F _ 5 L J &L
W #$ 1 5 a & ! = 5 ! 5 5_7 5 5n
3 fI 0 ¤
( 7A@ B
/ ¢ 3 m fI <3 ¢ ¤
E
E
/ ¢ 3 f 0fI 3 1 0 fI m 1 0 fI H fI <3
fI H fI 0 ¤ 1
/ ¢ 3 f 0 fI 3 1 fI 0 m 3 1 fI L fI <3 0 fI L1 fI 0 ¤
4.22.12. Integrals containing E (z) and Lλ n (z) 1.
1 -21 c 0f E / ¢ 3E
( ' 5_7 m !9 (P 5_7X! ' N ' % 5_7X! 5 1 L & & 5_7 5 1 ! 5 & J
Z
4.22.13. Integrals containing products of E (z) and K (z) 1.
1
Z
7 (
K
E
N
( 1 1 1 1 1 1 #$ ¤
1 # 7V7 #7 #7 7 !
,9P :
=
@CB
2.
1 -21
Z
/ ¢ 3
! 1 1 5_7! 5 1 L N & 7 5 J
K f;/
E
E
1 -21
K
/ ¢ 3
m
_ 5 7 ! 1 5
&
#$
7( m B ! ( #$ ! m 1 1 5 E 0f / E
N & 7 5 5 ! 1 1 1 J L 7 ( B m
Z
E
3.
!
4.
1
Z
1
Z
K
/ ¢ 3 E 0f;/ E
f 7 ( m ,77 5( m 0 fI 3 3PfI m m
/ ¢ 3
7 7 ( m 7<7 5( m 0fI 3 43fI m m m m
Z1 ¤ ¢ 3 7 6. K / E E / Z1 ¢ 7 ¤ 7. K / E E / 3
7 ¤ Z1 ¢ 7 7X
K / E E / 3
8. ! " Z1 ¢ 5 3
9. 7 ( m E f / E K / 1L
#$ J m Y N 7 1 & 5 ! 5 & 1 1
7 Z1 ¢ 3 7,75( m 10. 7 ( m E 0f;/ E K / m m 5.
K 0f;/
E
E
7( m
B
7( m
B
7 ( m
7 ( m
,9P<
!
B
B
+
11.
12.
"
5) , , 3
! ¢ K / 3
5 1L J
( #$ m & 1
7<5 m Y N 7 5 ! 5 ! & 1 1
7 Z1 m ¢ 3 f ¤ / G F E K <7 5 m 7<5 m m " ! 5_7X! Z1 m ¢ G / E 3
7<5 m K F 7,5 m
5 1L 5 &L J J
5_7 ( #$ m 1 1
7,5 m Y N 7 5 5 ! ! & 1 &
7 Z1 m ¢ 7<5 m K F 7,5 m G E / 3
3 0f9 3 ¢ fT3 AfI 3 3AfI m
m m 7<5 m
1
Z
m E F 7<5 m G
<7 5 m
13.
1
15.
Z
16.
Z
17.
Z
18.
Z
19.
Z
K
1 ¢ /
K
1
1
7
7<5
7<5
/ ¢
K
7,5
) *
,
/ ¢ 3
E
)+*
,
E
7,5
)+*
K
/ ¢ 3
1 ¢ -21 f 1
Y
K
E
G3
£ ¤
G
£ ¤
7<5
/ ¢ 3 £ 7,5
) *
( 1 N & 7
,
B
/ ¢ 3
B
] ¤
G
£
,
# ¤
G
! m 5 E F 7<5 m G
1 J L 1 ( m #$
7<5 m
5 1 ! 5 1 !
,9P L
/ ¢ 3
E
B
14.
B
E
4.22.14. Integrals containing products of E (z) 1.
2.
1
Z
E
7 (
1 -21
Z
E f;/
E
1
Z
1
4.
Z
5.
Z
6.
Z
7.
Z
( 1 V ( 1 1 1 1 1 #$ ¤
1 V7 #7 # 7 7 !
1
/ ¢ 3
( _ 5 7 m #$ ! _ 5 7X! 1 1 !
5 1 L 5 & L N & 7 5 1 5 & 7 ( J J
m !
/ ¢ 3
m 5 7 m ( m (:7 m 3 m m
E 0f;/
E
1
E
8.
3.
N
/ E
E 0/
2
E
E /
"
E
E E
E
E
/ ¢ 3
7 ( m 7<5 m 0 fI 3 4 3fI 7 ( m
7 ( m
B
/ ¢ 3
E f;/
E
E
¤
/ ¢ 3
¤ 7 7X ¤
! 5_7X!
5 1L 5 &L J
J
5_7 #$ m! 1 & 7 ( Y N 7 5 5 ! & 1 &
7 7<5 m Z1 7 ( m E f;/ E E / ¢ 3 m D0 f9 ¢ 7 ( m 3 £ f 7 ( 1
B
E
/ ¢ 3
,9P P
m
B
H
m
B
# #
9.
5) ,931
1 ¢ m f 1 E / ¢ 3 E F 7,5 m G
5 m 5_7 7 m 7 3 g f 3 m AfI 3 4 3 AfI m m 7 ( m B
Z
7 7<5 m 1
10.
Z
11.
Z
12.
Z
1 ¢ 1
1
m 7,5 m G
EF
E
¢ 1
7,5
) *
E
E
,
E
7<5
) *
/ ¢ 3 m f 0f9 3 ¢ f 7 ( m B
,
/ ¢ 3 £
E
¢ ¤
G
/ ¢ 3 £ £
G
]
¤
4.23. The Complete Elliptic Integral D (z) 4.23.1. Integrals containing D (z) and elementary functions 1.
Z
-21 0fg3 -21
D
W W & 1 & f 4 a 2- 1 N & 9 a 9 ! a a 1
2.
Z
3.
Z
4.
Z
D
D
fg3
0fg3
£ m F 3
!
fg3
F £ 3
3 f G h
( m
( m
( m
3hf G
1 0fg3 1 D fg3
m m m , 3 ) , 1)
#$
B
B
B
3 N N
m
, & ) ( m
B
5.
Z
-21 fQ3 1
D
0f 3
m £
1)
D
m
, 3
) ( m
6.
Z
-21 fQ3 -21
D
fQ3
£
1)
m
, 3
) ( m
7.
8.
9.
4a 1 0fg3
Z
Z
1
Z
D
-21
D
£ - !- & &4
f9 4a &4
fg3 W #$ ] 5n ( 7 Y N 1 & 5 !
! &
m m fg3 F #" G 3 *
D
fg3
Z
Y 11.
12.
£ -
f 0fg3 m m m 1) , 3 ) , & ) ,
-21 fT3 1
Z
D
-!&4 fT3 -21
SZ
V-
B
m
,
B
n 5 ! 5 L J
( m B ¢ 3 m ( m
B
( m
B
0fg3
£ -!&4 £ m 3 m , ,
) 1) m ¢ 1 S D
3 #- & N 7 (
D
,
-21
10.
f 1 ¢ ¢ 3 m *
m
( m
( m
B
3 N1
B
1 & #$ # 7,! 5 3 ¢
7 ( B
# #
Z
13.
D
-
( ( 1 1 m N & (
5) ,931 , 1 !
m
#$
3 ¢
!
7 (
m L -21 D
9 a J 1 L J m! #$ 7( 1 & Y N Ia 5_7 m m ! & 1
5m m ( Z m
D
0f 7( m m ! Z m 5 m (
D
¢ 3 ¢ 3hf 7<5 7 ( m )
, 7( m m ! 9a 1 m 9a Z1 -21 D f J 1 L N 1 & 5_ 7 5_! 7
& 7 ( J L m m 5 m (
Z
14.
15.
16.
B
B
B
B
B
B
B
B
17.
!
1
Z
18.
m F f
D f
f
¢ 3hf 3 ¢ G
7 ( m
4.23.2. Integrals containing products of D (z), K (z) and E (z) 1.
1 -21
Z
K
/ ¢ 3
D f
/ E
#$ ! 1 & ! m 5 N
1L & 5 1 5 1 J 7 ( m
!
2.
1
Z
K
/ ¢ 3
D f;/
E m D ¢ h 3 f f 3 N,
7<5 m 7 ( m H
7 ( m
1
3.
Z
4.
Z
K
/ ¢ 3
1 -21
E
E
D 0/
/ ¢ 3
D f
£¤
! 5 7 ! 1 &
5 1 L 5 & L N & 5 J J
_ 5 7 ! 1 5
5.
1
Z
/ ¢ 3
E
E
D f;/
/ E
f m (:7 ,77 5( m m , m )
m & !
#$
7( m
7 ( m
1
6.
Z
7.
Z
1
E
/ ¢ 3
E
/ ¢ 3
D 0/
B
B
E ¤
D 0/
E
¤
4.24. The Generalized Hypergeometric Function p Fq ((ap ); (bq ); z) 4.24.1. Integrals containing p Fq ((ap ); (bq ); z) and algebraic functions 1.
2.
-21 0fg3 -21 N a m ! ! ! m ( ! ` 1^ W m ! #$ Y a N a ! 9a I! a a m
1
Z -21 0fg3 -21 N a m ! ! ! m ( 1^ #$ ! m ! Y a N a ! 5
1 1 1 3 N 3
Z
!
f 4 a 2- 1
`
=m
!
( m
" " ! m 5 1L J
( m !
B
!
@ B
# #
5) ,!5) ,
4.24.2. Integrals containing p Fq ((ap ); (bq ); z) and trigonometric functions 1.
2.
3.
4.
m ! ! ! N ^ `
m ! 1 V7 m f £ £ a N a ! ! # 7 ( # < 7 5
m
f
! Z
m ! m £ N ! `
^ ' !' ' ( & m ! m a N a m !95:' ]'65 1 ! m ' % ! ' 1 1 !95n' O'65_
7 m ! Z
£ ¢ ! ! N `
^ m !' " ']" m !95n' ]'65 & ' % 3 O ! ' a 1 N a 1 !95n' O'65n ! m ! Z f ! ! N `
0f ^ " 5_7! ££ f 0f - 5_7 L a 5_7 L J ! a 5_7 #$ J
Y a N a ! m 5_1 7 a ! 5_7 = ( 1 -
! Z m ! m ! N ^ `
! " ' ' m !95 a 1 m ( m ! ' m ' N !95 '6 5_! 7 !' 5_7 L J
m ! Z
! ! f N `
^ m ! a 1 5_7 #$ " " 5_7X! ! a 1 N a ! - 5_7 a 5 7 a 5_7 L - 5_7 L J
J
= ( SZ
0f f
¤
¤
¤
5.
6.
7A@ B
¤
3 N95
7R@CB
7.
SZ
-
m ! N ! ! `
^ "( 7( a N a m
1
#
.
. W + + a 1 #7 ! \ U $ . .
+ + a 1 #7 ( #7<5
W m R
.
R. a ( " + 5_7#7 \ U #$ 7( + 1 ! m 5_7 a 1 N a + . a 1 + . 5_7 & ( & 5
= m @CB SZ m ! ! ! 8. V- N `
^ "( m ! 1 #7 #$ 7( a N a ! ¤ m
! #7( V7<5
SZ m ! ! ! N 9. - `
^ ( "( m ! V7 & #$ 7 ( ( 7X ! 5_ 7 a N a ! ¤ m
! &<- & a
m ! Z ! ! `
10. - N ^ W
R.
R. a #7 \ U #$ 7 + + 1 ! m a 1 N a + . + . a 1 #7( V7<5
m #$
. a
. 5_7 #7 W \
U + + 1 ! = m @CB a N .
.
m 5_7 a 1 + a 1 + 5_7 &<- & a
11.
Z
-
m ! m ! 7 ! ! N ^ `
m a N a ! 7( ! 3 N;
#$ 1 V7 !
V7<5
m =
@CB
# #
Z
12.
-
5) ,!5) 3
m ! ! ! N ^ `
m ! #7 & 7 a N a ! m 5_7 ! & - & a
#$
@CB
= m
4.24.3. Integrals containing p Fq ((ap ); (bq ); z) and the logarithmic function 1.
Z
-21
m 5 m (
m ! N ^ ! ! `
m ! ! f
5 1 L a N a ! 5 J
m ! 1 5 7
=m
@CB
4.24.4. Integrals containing p Fq ((ap ); (bq ); z), K (z) and E (z) 1.
m ! m ! ! `
m ! m ! a N a
5 1 L ! 5 1 ! 5 J
! Z1 m m 2- 1 E / ¢ 3 N ^ ! ! `
m ! 5_7 ! 5_7X! ! a N a
5 1 L 5 & L ! 5 1 5 J J
1 -21
Z
K
/ ¢ 3
N ^
1
=
&
=
2.
m
@ B
@CB
4.24.5. Integrals containing products of p Fq ((ap ); (bq ); z) 1.
2.
-21 0fg3 -21 N ! L N ! m ( ! `
1 1J 1 1^ f 4a -21 N 5 !5 m L 1 1J
Z
-21 0fg3 -21 N ! L N ! ( m ( ! `
1 1J 1 1^ W W 4f 4a -21 N ! 1 Ia 9a a 1
=m
@CB
Z
3 N :
#$
=m
@ B
3.
-21 0fg3 -21 N ! L N ! ( m ( ! `
1 J 1 ^ W W #$ ( f 4a -21 N ! =m
a 1 9a 1 9a a 1
Z a a2S -21 0fg 3 &-21 N 1 K3 N 1 3 fg3 4
. 3 ¢ SEa ; F 5 m G + a a2SEa &-21
S . ' Y L $L ¢ 383 S ¢ 3 )3 F 5 m G + a
& J J £ f2 = m Y a a a a2SEa &-2 1 #$ ( Z . 1 E S a ! a 1 0fg 3 + -21 N 1 3 ] fg 3 4 1 N '65 (*' & 1
7 ' 7 3 ¢ F G R ¢ 3 S $ L F G J &Y S? a a 1 + 1A- .
. / f / / f + -21
Y m . + -21 / f / 3 / f = #$ ( V( ( #$ (' V(' V(')( ( 1 Z 1 ! !
- S -;&- N
N
1 ( V( (: 7 1 (*' ( O')(: 7 1 43 ¢ SE a # % ! ! % ' % '! ! % D C £ £ F 7 GIH = m ! ( Z 2 1 fg 3 3 N a 1 ^ !&! ! ` ¡ N 1 f 5 m ( !& L ( m ! ! J Z
4.
5.
@CB
@CB
6.
7.
3 N<
@ B
'I@ B
7 B
Chapter 5
Finite Sums 5.1. The Psi Function ψ(z) 5.1.1. Sums containing ψ(k + a) 1.
2.
3.
1 fI fI
¢ <3 <¤ fT3 ¢ ¢ 3 0fg3 ¢ 0&3 0f 2 ¢ 3 0f 0 ¤ 7 £ m (8 3
f £ f fI m 5n ' fg3 3 £ ¢ 3hf f- ¢ 0 ¤
' fI ¢ ¢L£ f ¢ m (:7! 7 3 fI 0fg3 ¢ '6 5_ 7 ¢L£ f & 3 5. & fI £ £ £ R 4fg3 4.
6.
7.
m (87! m
$ " ¢ £ ¢ 3 1
3hfI <3 f2 £ fg3
3
f- ¢ ¢ X £ ¢ f- ¤
# f9
7 £ ¢ R ] 0 fT3 4
f9 fI 7 ¢ 4 3 0fg3 ¢
f9 R f- ¢ ¤
C
£ 3 £ -; £ "' 6 ' 3 5n
7 #7 ( 7 ' 5 ! L = '8>?7A@ B N 1 J 6 £ f 3 £ If £ f 0fI 3 £ ;- £ 0f 0f$ ¤ - T 1 3 N P
+
¢ '$ L 3 J 1 m !" ¢ 9. $&% !" 10. $&% O m !" ! 11. " fI ( 7 5_7X! K3 ¢ m
F £ 3 '7 G
8.
7
m
C
;
7 F¢ 3 ' G
= '8>?7A@CB
¢
m 5 7 ! ' ' % m f ¢ <3 ¢ ¤
5_7X! ' ' % O 3 ¢
¢ ¤
m m 5 7! ' m m 5_7! ' ! ' 0 fg3 ¢ D ¢ ! ' H fI 3 ( m (:7X!A m 5 7! ' !' f f ¢ h 3 f 0f ¢ < 3 ¢ ¤
12.
13.
14.
15.
16.
3
m (:7 ¢ !" m ! " fI ( m _ 5 7! fg3K3 ! '%& 3 ( m 5_7! m ! ' m ')( 9 $ %
¢ fg3 ¢ 0
¢ 0fg3 3 ¢ 0f$ ¤
!" 5 7X! fI $ _ m !' & '65_7X! % 0 f ¢ 2 3Pfg3 <3 ¢ h 3 fI ¢ ¤ 3hfI 3 4 3fg3
£ ' £ If $ L 3 I f J 7 £ &-21 F m 5_ G ( 7! % O')( $ : ')( $ ! % 1 ']" 1 L ' ' J D£
£ 3
C3
D£
7 '
£
m F G
m 5_7 F GIH ¤
7 F G 3 £ H
£ ')( $ ! % ¢ £ ')( $ ! % &-21 ¢ 3 31N
C
¤
= '*>?7R@CB
17.
' O ' -21 m (:7! " $9% fI J $ L J $ L " ' 5 7 m ! ' D £ If <3 f 3 F m _ G 1 L ' m !' J
F
m 5:'g5 7 ¤ GIH
5.1.2. Sums containing products of ψ(k + a) 1.
2.
3.
£ £ g ¢ fI ¢ h fI E f 3 3 fI E fI ¢ £ 3 £ If fI fI E fI ¤ 3
!" m ! " fI O m (:7 m ( (87! m ( (:7 O fg3 ! '%& 3 m ! ' m ( (87! D m ( (:7 ¢ f- ;
f 3 & I ¢ £ 3hfI & 0 fI
¢ ¢ 0 fg3 3 ¢ O ¢
fg3K3 ¢ ' ¢ H ¤
¢ 3 £ fI 0fI £ g f 3 ¢ E 0fI £ fg3 ¢ fI ¢ 3 £ 3 £ fI E If 7 7 ¤ fI & If fI <3 fI
5.1.3. Sums containing ψ (k + a, z) 1.
¢ ' 4 3 J $L 1
2.
( 7 ! " ' _ 5 7 $L $ J 1
F 7 G 3 ')(:7! % D C £ £ 1 J L'
F 7 IG H = '*>?7R@CB
F 7 G ' % 3 &L' J
' 7 ( ' V7#7 #7 N & ] ! 7 &L ' &
31 J
= '*>?7R@CB
+
3.
0 fI 3 7 E 0fI 3 If fI I f 7 7 £ fI T f 3 ¢ £ ¢ 3hfI If 0 0fI 7 ¢ £ £ 3 f £ fI fI 0 3
; ,
fI ¤
5.2. The Incomplete Gamma Functions γ(ν, z) and Γ(ν, z) 5.2.1. Sums containing γ(nk + ν, z) 1.
2.
3.
4.
5.
6.
7.
8.
")" " 7 #7 1 5 $ (:7 & (:7 N ]! L J
7 #7 " ! 7 L ¤ 3 5n' N J 5n'65_ ( 1;- #7 ( 7 ! " !¢ 7<5h ( X7 ! ' ( ! ' ¤ 3 ' % -&N
" &$ % & ' % " '65_7 ¤ ' 1 ! ' & 1 N 1 J 5:'g!5 7 L J $ L 43& - & 7 ¢ ' ! " $ 5 7! $ 4 3 L & J 'g5 7 7 D T3 ¢ )3 ¢! & <3 ' ! % -21 - -21 0& H ¤ ' " m (:7X! " 7 (*' m ¤ ¢ ' & ! ' L m ( '! " ! " $ 4 3 L N J 5_7 m (* J ( 7X! " ' ¢) ¢ S ¤ S 5 ! $ % J $ L & 9l 3 3 - SE- a -21 0& 7 ' ¢! ¢) ¢) 7IGIH ¤ $ L & ;G 3 F 1 D F J ' % ' $ 5n'65_7! % $ 5_7! % 43& - ¢) & '65_7 $ L J ' % ! 'g5: ¤ a
6 ' n 5
! % - 1 N 1 J O'65 ! L 3 31!,
γ (ν, z )
Γ(ν, z )
( ' m #7 ' m !" ! " 3P& - !¢ & & N 1 " ` $ L J ^ ! ( ' m V7 ( " ¤ ( m !' 3 ! ' - & N 1 ^ m ( (*! '65_7 ` m 5 ! ( ' m 5 !" ¢! 2 a S 5 ! 5 5n'65_7 L 10. J $ L $ % 3P& - & 3P& Jm ( (*' m #7 C7( m ! ' 3 5:'! % S V- & N 1 ^ m ( ' ( " ` ¤ ! ' n 5 ' ! " ¢ £) & 5_7! " ! " $ 4 3 L 11. J " '8>?7A@CB 5_ 7')! (: 7X! 5_% ' 7% ! ' ']" 3P& & -21 0& = 9.
5.2.2. Sums containing products of γ(ν
k, z)
' $ L 1 & - £ 3 1 3 & J * W #$ '& ' '65 1 '65_7 5n' £ ! ¤ ! ' & 5n'! & N a 1 5n' 5:'g5 7 5n'6 5 7 1
a 1 O'65_7 £ ¢ 2. & 3 3& nJ $ L W #$ '& 8 '& '%& '65 & '65n 5n'65_7 3 ¤ £ £ ! 5n'65_7! N ! '& & a & 5n' 5n'65n 5n'65n &
5.2.3. Sums containing Γ(ν k, z) ' 1 & - 77( ( 5n ' L ¤ $ L 4 3 & T3 1. ! J J ' ¢ 1 ¢ ¢ '8>?7A@CB 2. J $ L 3 ; T3 & 3 &-21 3 -; V- & -;-2 1 & = C7 ( ! " ' % a '65_7 ¤ ' C 7 ( * ( ' ! ! 3. & " 3& T3 ' V- J 5:'g!5 7 L J $ L ')(:7X! % ¢ ' 7 ! " 1 & 3 ! ' - & -21 & $ 4 3 L = '8>?7A@CB 4. J 1.
313
+
; 31
5.3. The Bessel Function Jν (z) 5.3.1. Sums containing Jν
nk (z)
1.
( &$'% ! " £ & - a 0& * -; 1
2.
( &$'% ! " 43 £ & - 3 ¢ & * &- -21
3.
!" ' £) n 5 ' 6 _ 5 X 7 ! $ L a a " 0& 1 F G J ;
4.
!" ¢ ' £) ' 5_7! " $ 4 3 L ; 5n6 J
FO$
' ¤ G
' ¤ G
F
a 0 & ¤
a
& W " " 5 '65 1 ( #$ n ¤ ! ! 1 N 5 ] 5n
7
6 ' _ 5 1
5.
"
( L ( L J J (*'65_7X! ( ! ' J $ L " a 0 & '
'
¤ -; &
5.3.2. Sums containing products of Jν 1.
2.
3.
¢ 4 3 ; a & 1 ( 7! ' a
nk (z)
7
£ ¢
& <3 - a 1 0& ¢ a a 0 & 3h a a 0& ¤ 1
5 ( ! " ' L 1 ¤ $ L 5n'65_7X! " a 0 & J ! 1 N 5 7 ! 5n'65_7 J
a 1 0 & a a 1 0& £ -
a 1 £ )£ ¢ '65_7 3 J $ L ;- &-21 Q ;- &-21 0 & ' & 5 a n 3 ¤
(')( 1 L F G 1 -;&-21 & J 315
5.4. The Modified Bessel Function Iν (z) 5.4.1. Sums containing Iν
"
1.
2.
3.
4.
nk (z)
'
( L L ' J J (*'6 5_7X! " a 0& ( ! ' -; 0& ¤ $ L
J 7 ( $&'% ! " £ & - ¢ ¤ 0 & 43 - <1
£) ¤ ; a 0& a 1 & <3 a a 1 0& 1 !" ¢ ' £) ¤ $ 4 3 L 5:'g5 7! " a 0& ; a 1 F G a 0& J
5.4.2. Sums containing products of Jν 1.
' J $ L
-; a 0& (
nk (z)
and Iν
nk (z)
a &
#$ ( 7X! ' ( ! ' ¤ 5_X7 ! F G -; N & 5_7 - a 1 - 5_7
' 2. J $L -; a 0& - & #$ ' 5 1 ( ¤ ( 7 ! ( 5 7 ! ! ' F ,G -; N ! 5 7 1 a 1 5_7 ( a 1 ( _
' G 3. -; a 1 Q -21 & " J ']$ " L F ¤ A& a 1 1 -; A& 3 A& a 1 1 -; 3 A& ¢ ' G &- -21 Q -21 0& 4. 4" 3 ']" J $ L F ¤ A& a 1 &-21 A& 3 A& a 1 &-21 3 A& 3 ' & 8 ¢ ' ¢ 5. 43 J $L -21 & &- a 1 0& 3 '65 ! O'65 ! £ £ ¤ Y D a / a / H 1 1
31!;
+
6.
7.
8.
9.
10.
11.
12.
; 5) ,
3 '%& 8 ¢ ' 43 J $ L 2- 1 & -; a 1 0& '65 ! £ ' O'g5 ! Y D /
1 -;
1
£ ¤
-; / H
3 '%& 8 ' ¢ J $ L - -21 0& -; a 1 & 43 '65 ! O'65 ! £ Y D a / ' 1
a 1 / £ H ¤
3 '%& ¢ ' 4 3 J $L a 1 & -; a 1 0& O'65 ! £ Y D
-;&-21 /
8 '65 !
3 '%& ¢ ' 4 3 J $ L 2- 1 & ;- &-21 0& '65 ! £ Y D
-;& -21 / E3
8 '65 !
-;&-21
£ ¤
/ H
-;&-21
£ ¤
/ H
-;&-21
£ ¤
/ H
Y
3 '%& 8 J $L 1 - 0& &- a 1 & O'65 ! £ '65 ! /
D -;&-21
Y
3 '%& 8 J $ L - -21 0& & - -21 0& '65 ! ' 5 ! £ E3 6 £ ¤ / D
-;&-21
;- &-21 / H
'
'
3 ' & ¢ ' ¢ 43 J $ L a 1 & &- -21 0& 43 '65 ! O'65 £ Y D a / 3 1 31:
8 !
a 1 / £ H ¤
13.
14.
' J $ L 1 Y D ' J $L 1 Y D
3 '%& 8 £ K3 O'65 !
0 & &
&- -21 '65 !
1 -; / 0& -;&-21 & '65 ! a / 1
! " ¢ ' ' 5_7! " V a & $ L 4 3 - 5:6 J
4.
5.
6.
£ ¤ 1 -; / H
5 J L ! N 5_7 1 !5n'65_7 ¤ 1
65 (*'65_7X! ' a ' ¢ 6 _ 5 7 ! n 5 6 ' _ 5 X 7 A ! ( ) ( ! G $ L 0 & a & 3 F ' J a a a a a 5 7 1 ¤ Y N 6 _ 5 7 n 5 6 ' _ 5 7
g5 5_! 7
& ' J $ L - 0& - 0& a - a 1 a - 5_7 ( ! ' ( ! ' a 6 ¤
! 5_7! 5_7X! F G
N & )(*'65_ 7 (*'65_7 65 ( 'g5 7
a 1 ¢ £ £! ¢ '65_7 3 3 J $ L -;&-21 Q -;&-21 0& ' & 5 a 3 ¤
( ' ( 1 L F G 1 -;&-21 & J
" ' ( 7 ! G $ L V
3 V
D F F SEa 1 $ GIH $ - S -21 $ J 1 3 ¢ £ 1A- - &-21 = O' '% ! % @ S 3 3 ¢ S = 'I@ B l " ( 7! M ! '%& M ! " M ! "'& M ! 43 ¢ '%& M ]! M ]! ¤
3.
nk (z)
2.
3 '%& 8 ¢ 3 £ K3 O'65 ! a / £ H ¤ 1
5.4.3. Sums containing products of Iν 1.
31!<
+
; ;
5.5. The Macdonald Function Kν (z) 5.5.1. Sums containing Kν 1.
2.
3.
'65 ')( $&% 7
nk (z)
$ !% $ ! % a 1 & 7 a 1 ) 3 * 3 a , 1 ) ")" 7 $&% ( ' ( ! $ ! % ' % " V- ¤ &- -21 & 1 ( m (*' L)" J ')( $ ! % C7 ( m (* '! " F G a 1 & " '! F 7 G 5n Om '5n ! - - m m 5n O' 3 '65_7X! % m 5:'! £ &-21 O 1 - N
O
7
* 3
,
= @CB
2- 1 0& m 5n 'g5 7 #7 O m 5n O'65_7 O'6! 5n L ¤ J " "']" ' L ¢ E S a - S -;&-21 0& ¤
( ' )S - a 1 0& 3 5 1L' L" J
F G a & a & ¤
4.
5.
( ' J J $ L 1 ( J
' J $ L
5.5.2. Sums containing Kν 1.
¢ ' 43 J $L
nk (z)
and special functions
a -; & a 0 & ( L J
F3 G N & E - 5_7 a 5 a 5 7L J
( a E Q 1 L 3 J a a F3 G N & E- a & a a & L J
) ' ( ! ¢ 3 5_7X! F ,G -; N & - a 1
#$
7 1
#$
(
31L
& &
- 5_7 5_7
#$
¤
2.
3.
2.
Lν (z )
¢ ' G <1 Q 2- 1 0& 43 J $ L F " ']" 3 ¢ a Q3 Q 1 & -21 Q3 Q Q a 1 &-21 Q 0 ¤ ¢ 0& 3 0& a 4 3 & -21 0& 3 a 1 & & 0 - ¤ ¢ 3 a & &- a 1 & <3 a a 1 0& &- 0&
5.5.3. Sums containing products of Kν 1.
Hν (z )
nk (z)
)£ ¢ a 1 Q a 1 0& 5 a $ 5:'g5 7! % ¢ / $ 5_ 7X! % ')( $ ! % F G 1 a 1 & = @CB $ 5_7 ¢ ')( $ ! % $ 5n'65_7X! % a 1 Q a 1 & 4 3 43 ¢ ' % F 5 G a 1 a & ¤ 1
5.6. The Struve Functions Hν (z) and Lν (z) 5.6.1. Sums containing Hk+ν (z) and Lk+ν (z) 1.
2.
3.
( ' ! " '! " &$ % F G H -21 & 3 ¢ " a F ,G D £ £ 1
( ' ! ' ! " " $&% F G L -21 0& " a F G D £ £ 1
' F]3 G H a & $ L 0 I f J " '& ( m 5 &L ' J 5:'g5 & L J
31P
¢ a F , G 3h a F , GIH ¤ 1
&4
¢ a F G a F GIH ¤ 1
&4
( m 5:'g5
N & 5:'g5 &
W & V7 ( #$ ¤ ! ( m 5 & &
+
4.
' G L a & J $ L 0fI F]3 " '&
( m 5 &L' 5 '65 & #7 ( m n ! J :
N 5 'g5 & L & 5:'g5 & ( m 5 J
; <
W
& &
#$
¤
5.7. The Legendre Polynomials Pn (z) 5.7.1. Sums containing Pm 1.
2.
3.
nk (z)
( ' ( ' (')( #$ 1! ¤ O'! % E (:7 G &N
j
1;- & ' % ! F 1 (*' V( O')(:7
£) ¢ '65_7 (:7 j a 1 & <3 j & ¤ j & 7,5 4 ' ¢ £ j & $ j F 7,5: 4 5 G ¤ $ L J $ ')( $ ! £ 7 '8>?7A@CB ')( $ ! % & j & F G £ & = 1 " ( 7 ! ')( $ ! % m ! " FO63 3 ¢ G j & 7 ! j + !- &4 lM1A- -; . F 3 # V 3 ¢ G ¤
m ' & -21 ¢ ' ¢ G j 0& F $ L 3 3 3
J ' 1L' ('! " J ' % F ¢ 3 (87 G -; £ - C7 (*'! " F (:7 3 ¢ G ¤ ' m !" 7( G j & J $ L $&% F ( ' #( ' #$ 19- <- ]7,( a ¤ m ' ! % ' F 727,5: ( G N & 7,( m ( ' 7( m ( ' 7
!
4.
5.
6.
7.
8.
3,9N
& '
)" " ' 9. J $ L ')( $ ! % 1 (*' L " ) 7 J
" ' F ¢ 3h 1 J L' m !" ')( $ ! % $ 5n'65_7! % ( 10. m !' ' ! O'Q 5_ 7!A % 11.
12.
13.
14.
15.
16.
8 ( 7 8 ( (8 5 # (:7 , j & #$ ( 1;- 1 ¢ ¤
3 G ;- & N 7 #7 7 (8 5 (:7 !
m ! " j & #$ ( ' ( ' ( m 5_7 & ! a
( m !' F G & N
1 ¤ 1 (*' V7 ( m ( '
m !" !" £! ¢ ')( $ ! % $ 5n'65_7X! % ( m ! " ( ! " j & ( ' ( ' ] ( m (
' ! m ( ! ' ! ! ' ( ! F 7<5 G N ¤ 7( m (*' V7 ( ! (*1 ' a % ' m ' &
¢ ' ¢ 3h G j 0& $ L 3 4 3 F
J ¢ 3h 3 ¢ j + 9lC-;&-21 . F £ 3 £ 3 ¢ 3 ¢ G ¤
¢ ' m !" j 0& $ L
3 J 1L" J
( ' ( ' &- - #$ ! '
; 1 m W ¤ - ¢ 3h
N & ( ( ' V 7 ( ( ' m m W 1 1 1 J L '
! -Q1 ' m !" ¢ - j & $ L h 3
J 1 J L" (' 1 (*' 1- - #7( a " ¤ m ! ' ¢ 3h
-; N 7 ( * ( ' # 7 ( * ( ' V 7 (
m m & 1L' ! J
$ 5_7 O'6' 5 % 7 ¤ j 0 & ')( $ ! % '65 & )L "
J
1L'& ¢ ' ¢ J 1 L)" J ' % ¢ 3h
¤ j 0& $ L 4 3 '65
J &L " J
3,
+
17.
18.
$ 5_7!& '! " 7 ( ' ' % 0& ¤ $ ! % & (*' L " & 5n' L " j & J
J
m !" - j & 1 J L " ( ' ( ' &<- - #$ ! '
; 1 m ' % - ¢ 3h
N W ¤ & 1 ( m (*' V7 ( m ( ' 1 ! W
-Q1 ' m !" ¢ - j J $ L 1 L " 3h O & J
(' 1 (*' 1- - #7( a " ¤ m ! ' ¢ 3h -; N 7 ( * ( ' # 7 ( * ( ' V 7 (
m m & 1L' ! J $ 5_ 7 ')( $ ! % $ (:7X!A $ 5 ! 'g5 & L " j & J
5_7 (' (*' V7 " #$ O 6 ' 1 ' % ')(87!A '65 ! N ¤ !
* ( &
1 ' ( ' $ 58'! % 1 L " ' 43 ¢ J $L ¢ $9% ( ' J 58' j & 1 & J L"J L" £ ¢ I j 0& ¤ ' _ 5 7 $ 1 " L j 0& 43 ¢ J $L $ (:7X!A $ 5 ! '6J 5
& )L "
J
(' V(' V7 #$ & 1 W ¤ L ' ! ' % ')(:J 7X!A '65 ! ¢ 3h O N 1- &
1 (*' (*' m !" !" 1 L " ' J
j 0& 43 ¢ J $ L ¢ '65 & L)" & ( m L" & ( L"
J ] m ! ' ! ' 1 L J ' & J
( ' ( ' & ( m ( 1 W ¢ ¤ J 7( m (* ' V7 ( ! (*1;' - ' % & ( m L ' & ( L ' 3h & N
J
J
$ 5 O'6' 5 % a 1 ¤ j a 0 & ')( $ ! % '65 L " 1 J
¢ 43 ')( ¢ ' 43 J $ L
19.
20.
21.
22.
23.
24.
; <
3, ,
25.
26.
27.
28.
29.
& '
' m !" ¢ - j a & $ L h 3
1 J & J L" (' V( 1 (*' 1- - #7 ( a " ¤ m ! ' a 1 ¢ 3h
-; N 7 ( * ( ' V 7 ( ( ' V 7 (
m m & &L ' ! J
¢ ' m !" j a 1 0& $ L 3
J & L " J
( ' ( ' - - #$ ! '
& m ' % 1A- ¢ 3h N W ¤ & 7 ( m (*' & ( m ( ' & ! W
-Q1 ' ! " m ¢ $ L 3h - j a 1 & J &L " J
(' V(')( 1 1- - #7 ( a " ¤ m ! ' a 1 ¢ 3h -; N 7 ( * ( ' V 7 ( ( ' V 7 (
m m & &L ' ! J
& L '%& &L " ¢ ' £) j a 1 & J ' % ; ¢ 3h ¤ $ L 4 3 F G '6J 5
J L" J
7 $ 5 ')( $ ! % $ 5_7X!A $ 5 ! '65 j a & L" 1 J
( ' ( ( ' V7 " #$
6 ' 5 1 ' % '65_7X!A '6
¤ ! a 5 O! 1 & N
( 1 ( ' & (*'
30.
31.
&L " ¢ ' $ 5 $ 5_7X!A $ 5 ! '6J 5
j a & $ L 4 3 J L" 1 & L ' &J
( ' V(' V7 1 W #$ ! ' % '6J 5_ 7X!A '65 O! ¢ 3h N ¤ ( 1 (*' & 1(*- ' &
! ! " " m & ¢ ' J L " ( j a 0& $ L 4 3 '65 ( J L" mL" L" 1 J J
] m ! ' ! ' & L ' & J
( ' ( ' ( m ( 1 W ¢ ¤ J
7( m (* ' V7 ( ! (*1;' - ' % ( m L ' ( L ' ; 3h & N
J
J
3,93
+
32.
33.
34.
35.
; <
( ')( 1 ¢ $ &L £ 3 ¢ J ( '65
* J J £ SEa 1 ' ( ' % ! % F 7 G ]F 3 3 7 S
L" j & 1 L " & -
S G SEa ¢ h 3 & SE- a 1 S & 1 = ( ' ( ' 43 ¢ £ 3 ¢ J $&% 1 L" j & F G ' %! ¤ &-
* ( ' ! ( " £ 3 ¢ m J 1 L" j 0& J & L ' a 1 0& ¤ $&% 1 ( m (*' L " &-
m 5 1 J L' J
U0\ U0\ & O')( $9 % $ 5_7 F3 O'65_7 G j 0& ] j l F E(:7 G ¤ &!- & ('! " ( R a 1 L " & 3 ¢ £! ¢ $9% ( J 'g5 1 L " j &- 0& J
( ' ( ' ( a 1 R & W ! £ J O& 'L ! '% 3 ¢ N ( ' ( 1 V( 'g5 <1 - &
( ' ! ( a " R &L " & 3 ¢ £! a 0& J' ( j ( 9 $ % 1 L " &- 1 J
( ' ( ' ( a & & L R & W ! £ a 1 J O '65_' 7X& ! % ; 3 ¢ N , (')( & V(')( 1 &
¢ ' ( ! " ¢ ! 3 G j S - & 43 J $L m " F m 5 ! ! ' F 3 ¢ G S
m ' ( #7( m ( 7 (8 5n (:7 1 Y N 7( m ( ! (*' V7 = > &
5_7X! " ¢ ' ¢ G j a2S 0& 3 43 J $ L (*'65 & L " FO$ J
1L' S 3 ¢ G - - ( J ( 1 L ' F J
Y j S + lC- S -;&-21 . V V 3 ¢ 3 43 3
'I@CB
36.
#$
¤
37.
#$
¤
38.
39.
3,!5
'I@ B
¤
& '
5.7.2. Sums containing Pn (z) and special functions 1.
7 ' ' '#" " (8 $L O ]! " & - j 0& O ]! '#" j &-21 & J
= '8>?7A@CB
2.
O '65 7 '65_7 £ $ 5_7 L& & - &- j a 1 & ' '#" 1A- j & ¤ J
3.
F Q3
¢ G 3
¢ j & £ j + 1RlC-;&!- &4 . F V 3 3 # 3 ¢ G
. ¢ ¢ ¢ F 3h 3 G -21 D 3 j + a lC-;&- F # 3 3 # 1
5.7.3. Sums containing products of Pm 1.
2.
3.
4.
3 ¢ G9H ¤
nk (z)
)£ ¢
j 2 & 0 'g5 7 ! j 0& 3 £ j & j a 0& j a & 0 ¤ 7 (* 1 1 $ 5_7 1L' ¢ ¤ ')( $ ! % $ 5n'65_7X! % j 0& J ' % ! ¢ h
3 4 3 $ 5_7! '65 1 L " 7 (* ' ¢ J 5_7X! % * j 0& ' % ! j & ¤ 4 3 ) ' ( ! n 5 6 ' ( ' $ % $ & J L" 7(8 7,5 ¢ ¢ J 1 L" 4 3 $9% j * , j * , ) ) 7 (8 7,5 £ 43 ¢ a 1 ¢ J & ' L % ' j a j a ,
) *
) * , ( ' ]'65 #$ L ' ¢ ¤ J ' % ! 3h
N & & 7 (8
! 3, ;
+
5.
¢ 43
; < 5
7 (8 7<5 a j ,
1) * ) * 7 (8 j a & * , j a & ) ) L' ¢ 7V J ' % ! 3h &4 N &
&L " J $9% j a 1 £ 43 ¢ a 1 ¢ J ' L % '
,
7,5 * , ( ' ]'65 #$ ¤ 7* (
!
5.7.4. Sums containing Pm (ϕ(k, z)) 1.
2.
3.
¢ ' j 43 J $ L S ¢ '$ L j S ¢ 4 3 J 1 ¢ ' $ L j SEa 3 J
= 'I@CB
& ¡ & 3 £ & S F 7 G S
S l 3 ¢
= '8>
@CB
&
' 5 6 £ F 3 G S J $ 5n' 1 L ! % " £ & a a 1 Q ¤ S a
4.
5.
6.
7.
8.
1 L ( ¢ ' j S / 3 ¢ S S J
$ S l 4 3 L J 1L( J
& L ( ¢ ' j SEa 1 / 3 ¢ S SEa 1 J
$ S l 4 3 L
2 1 J &L ( 1 J
&L ( 3 43 ¢ S J % ¢ ' $ L j S / ¢ 3P& S J 1 L ( 4 3
J 1 ( J L ( 7! " ' j SEa 1 / ¢ 3V& S $ L < 7 5 $ J 1 ¢ ' S j S:F ¢ $ G 43 ¢ S S l $ 4 3 L J 1 3,9:
= '8>
@CB
@ B
= '8>
@CB
% & L ( J 5_7X! % S l 3 ¢ = '8>
@CB
S l
= '8>
1L( 3 J % £ & S
= '8>
@CB
& '
¢ 4 3 1 10. 3 1 9.
11.
12.
13.
14.
15.
16.
17.
7 ' J $ L & S j S +) * $ 5 , F G S S l 3 S
= '8>
¢ ' $ L & SEa 1 j SEa 5 , / F G S S l $
1 )+* J ' > 3h SEa 1 = * '8> ¢ ' $ L 3h& S j S F $$ 5(8 G 3 ¢ S ? S 3h S 3 = l J 1 7 1 ( ¢ ' S J L ! % £ & S j G G $ L S S 4 3 F 3 = '8> F $ S l
J 1 ¢ ' SEa j $ L
E S a 4 3 F 1 1 $ G F G S S l J 1 & L ( 3 J 5_7! % £ S SEa 1 = '8> ¢ S ? 1 ( ¢ ' S ¢ J L ! % #& S G j $ $ 3 S 3 4 3 L S : # F "
l J 1 = '*> ( ¢ ' $ & j SEa 1 F " ¢ $ G 43 ¢ S _ S l $ L 4 3 n 5 $ J 1 & L ( 3 J 5_7X! % V& S = '8> $ ¢ ' 43 ¢ S _ $ 4 3 L & S j S * $ 5 , ) J 1 1 3 J
@CB
@CB
@B
@CB
@ B
@CB
@ B
= '8>
@CB
= '*>
@CB
S l L( % 3& S
$ ¢ ' & SEa 1 j SEa 5 , $ $ L
4 3 2 1
1) * J 1 3 ¢ S _ S 3 J & L % ( 43& S l 3, <
+
18.
19.
; < ;
¢ ' $ L S j S F $ 5 G 3 ¢ S F 7 G - S S 3 $ S l J 1 1L( 3 J % S = '8> ¢ ' S S j S ) 4 3 J $ L & 1 ¢ _i S ;- J 1 L(*( '"! ' 43 %
@CB
$ 5 3h S J 1 L % ( $ $ 5 ] ! , S -; ')( ! " ( ]! ")" ¤ 5n'! % ')( 5 a $ 1L" J
5.7.5. Sums containing Pk (ϕ(k, z)) 1.
2.
¢ ' $ L f f O &- -21 & j F $ 5 4 3 $ 5 J 1 ' 3 & -21 ' ( m 5 m
$ G ( m j F ( m G ¤
¢ ' ¢ f ¢
$ L 4 3 f &- 2- 1 J ( ]! ' $ m 5n 5n 1 L ' Y j $ 5_7!A $ 5 5_7! , J ' % ' m _ 5 7! ¤ m m )
5.7.6. Sums containing products of Pm (ϕ(k, z)) 1.
2.
3.
¢ ' j Sg $ 4 3 & L J
&
=
¡
¢ ' S j S:F#" ¢ $ " $ 4 3 L J 1 _ 3
G j S:F#" ¢ $ 3 " $ G $ T ¢ S S 3 J 1 L ! ( 3#& S = '*> % l ¢ ' S j ES a 1 F#" ¢ $ " $ GTj SEa 1 #F " ¢ $ 3 " $ G $ L 4 3 J 1 _ 3 ¢ S S 3 J & L ! ( 3#& S = '*> % l 3,9L
'I@CB
@CB
@CB
) ' * '
5.8. The Chebyshev Polynomials Tn (z) and Un (z) 5.8.1. Sums containing Tm+nk (z) 1.
2.
3.
4.
5.
6.
'
7,5 4 ¢ £ 0& F 7<5n 4 5 G ¤ m !" F 7(8 G & 1L" J
(' ( ' 1;- - 1 m ! ' F 7<7 5(8 G N & 1 ( m ( ' V7 ( m ( ' 1L' J
7 7 0& 0& ¤ 7 3 O & M (:7! £ 0 3 ¢ £ 0& 3h
$J L ' J $L
O' 7 O' ')( $ L & £ &-21 ' L ¤ J J ' ! 7 7 ')( $ ! % '65 $ ! % 0& ] O'! % ] ' % ! ¤
' m !" 43 ¢ J $L - 0& 1 J L" ( ' ( ' ! ' 1 1;- - m ¢
m (*' #7( m (*'
- 3h N & ( 1 1 J L'
' m !" ¢ - & $ L h 3
8. J 1 J L " ( ' ( ' 1 1;- - m ! ' ¢ 3h -; N
( ( * ' # 7 ( m (*' m & 1 1L' J
7 &-21 0& 9. a 1 0& O'65_7 ')( $ L a 1 & £ a 1 ¤ 10. J 7.
3,9P
1
&<- - !
, a - 1
#$
¤
&-21 0 & = '*>?7R@CB
& - - #$ W ¤ 1 W
! -Q1 & - - #$ " ¤ 1! (
= '8>?7A@CB
+
11.
12.
13.
14.
15.
16.
17.
18.
19.
7 O' 5n'65_7 ) ' ( $ $ J 3 R
; L1
' '& L a & O '65 7 ¤
1
¢ a 0&
1 7 M : ( 7X! £ ¢ a 1 0& 3h & 0 ¤ 7 ' ' & ¤ ')( $ ! % $ 5n'65_7! % a 1 & O'65_7X! % $ 5_7 ' ¢ ¢ ')( $ ! % $ 5n'65_7X! % a 1 & '! % 3h ¤ 43 ' m !" ¢ - a 0& $ h 3 O
L 1 J & J L" #$ ( ' (')( 1 1;- - & - - " ¤ m ! ' a 1 ¢ 3h -; N ' & 1 ( m (*' #7( m (* 1! ( &L ' J
m !" ¢ ')( $ ! % $ 5_7X! % F G a 1 0& 3 ( ' (*' - - #$ ! 7 * ( '
1 & m O'! % 1A- W ¤ , N & 7 ( m ( ' & ( m (*' & W )
! -Q1 C7,5 m 5n'! " ¢ ')( $ ! % $ 5n'65_7X! % C7 ( m ( '! " a 1 & 4 3 ] 5n7 '!A ! 0& ¤ m m ' a 1 3 ¢ ' $L C7 ( m ! ( " '! & ] ' %! ' & m m " &-
J ' m ! ' ( 7! ' ¤ C7 ( * ( ' ! m
'
l
( ' 8 ( 7 & J $& & % L)" & j + -I & lC- & . F G &-!& ( L ' 7 ¤ & & l J & % 3 3 N
&
20.
21.
22.
23.
) ' * '
(*' " ( O! ' . 43 ¢ $&% OJ ') 1 (* $ L 5_ X7 ! &- a 1 & O '65_7Qj + 1 lC-;&-21 ¢ 3 # O ¤
¤ £ S J $ L SEa &- & & & O'65_7 £ $ 5_7 L & - &- a 1 0& J O' ']" " (:7 $ L O ]! " &- 0& J
O'65_7 ' '#" 0& ¤ ' ! ']" 1A- &-21 &
= '8>?7A@CB
5.8.2. Sums containing products of Tm+nk (z) 1.
2.
')( 1 L " ¢ J
43 ')( $ ! % $ 5:'! % J ] ' 7 ! % ¢ 43 a 43 ¢
( ' L " 0 &
& * 7
( 1 L ' 1 L ' j 0& 3h j &-21 & 0
J J
7(8 7<5 a , 1 ) * , 1) * '65_7X! 7(8 a 1 '65 a & ) * , a & ) * '65_7X!A '65n ! ¢ ¢ 3h
3 ] O'Q5 ! a & F 7 ¢ 3 N 7(8
= '8>?7A@ B
7,5 , G
(')(* '65_7 ¤ 1 1 ! 7(8
5.8.3. Sums containing Tn (ϕ(k, z)) 1.
2.
3.
¢ ' ¡ $ 4 3 L T S & J ' ¢ 3 J $L SEa & S £ &-21 ;3P& a ¢ ' $ L S S:F ¢ $ G 3 ¢ S _ S 3 £ S -21 4 3 l J 1 3 31
= 'I@CB
S
! " a ' S Q ¤ $ 5n = '8>
>?7A@CB
+
4.
5.
6.
7.
8.
9.
¢ ' / 43 J $ L S ¢ ' 43 J $ L -21
1
1
1
10.
11.
12.
13.
¢ S £ S SEa _
SEa 1 / 4 3 1 S l 3 43 ¢
5.8.4. Sums containing Um+nk (z) 1.
= '8>
>?7A@CB
¢ S £ = '*> @CB
' S $ L S F $ G ? S l 3 £ S 2- 1 S = '8> >?7A@B ' SEa _ $ L
E S a F 1 1 $ G £ ¢ S l 3 £ S SEa 1 = '*> @CB ' 43 ¢ J $ L S S:F" ¢ $ G 43 ¢ S _ S l 3 £ S -21 S = '*> >?7R@CB ( & ' $ ¢ 43 J $L $ 5n SEa 1 F#" ¢ $ G 3 ¢ S ? S l 3 #& S = '*> @CB ' = '8> @CB 43 ¢ J $L & S S ) * $ 5n , _ S l 3h S 1 ' ¢ E S a
5 1 43 J $ L &
SEa 1 )+* $ , 1 43 ¢ SEa 1 _ £ ¢ A/ S 3h SEa 1 = '8> @CB l ' $ 43 ¢ J $ L & S S ) * $ 5n , 3 ¢ S ? S l 3 3P& S 1 = '*> @CB $ ' 43 ¢ J $ L -21 & SEa 1 SEa 1 ) * $ 5 , 43 ¢ S ? S l 1 3 £ ¢ X3P& S = '8> @ B
¢ 4 3 J 1 ¢ 4 3 J 1
3 ¢ S ? £ S 2- 1 S S l
; L1 5
" ' 5 7 $ L & J $
7<5n 4 5 ' 7<5 ¤ F 7<5n 5 G '65 7 3 3,
2.
3.
4.
5.
6.
7.
8.
) ' * '
¢ ¢ O '65: £ ¢ ¢ ' ( $ L & R 3h& ¤ 4 3 ) J ']" $ 5_7 ¢ ')( $ ! % $ 5n'65n ! % & '65_7X! % ¢ 3h& ¤ 4 3 " m !" ')( $ ! % $ 5n ! % ¢ 3h& - 0& ( ' ( (*' '#" 1 &- - - - '65_ m 7X!! ' % F 7<7 5(8 G N 7( m (*' & ( m (*' & , - &
! a 1 m !" ')( $ ! % $ 5n ! % F 7(* G 0& ( ' ( ' ( '#" 1 &- - - - '65_ m 7X!! ' % F 7<7 5(8 G N 7( m (*' & ( m (*' & , - &
! a 1 $ 5_7 ')( $ ! % $ 5n'65: O! % $ 5 7!A $ 5 ! 0& " 7 (8 3
; & 2 1 a & L ' '65 ! 7(* 1 ) * J
! ! " " m ¢ ')( $ ! % $ 5:'g5: O! % ( m ! " ( ! " & ( ' ( ' ( ( ']" m ( 1 '65_ 7! % m( ! ' ! ! ' ( ! ¢ & N m ' ' & 7( m (*' #7( (*' !
1a ¢ ¢ £ ¤ 3 43 0& #$
¤
#$
¤
¤ ,
#$
¤
¢ £! ¢ 4 3 0 & 3 O 7 3 ¢ ¢ a 0& 3 # ' 5h ! % - j + -!&4 lC-21 . ¢ 3 £
¤
a
&L ' J
' 5 7 ! $ ')( $ ! % $ 5n'65_7! % 0& '! % ¤ 10. 9.
3 3 3
11.
12.
13.
14.
15.
16.
17.
18.
19.
')(
+
m !" $ ! % $ 5_7X! % F 7 (8 G & ( ' m !' £ ¢ O'! % & 4 3h
-; N & 7 (
; L1 5
( ' &- - - - #$ 1 * ¤ m (*' & ( m (*' & ! ( "
¢ ' m !" $ L 4 3
& J &L " J
( ' ( ' ( # ! ' 1 1;- - &- W - $ ¤ m ¢ - 3h N & ( m (* ' #7( m (*' 1 W &L ' 1 J
'%&
! -Q1 ' $ 5_7 ')( $ ! % $ 5n'65n O! % a 1 0& O'65 7! % ¤ ' ¢ $ 5_7 ¢ ')( $ ! % $ 5n'65n ! % a 1 & O'65_7X! % 3h
¤ 4 3 m !" ')( $ ! % $ 5n ! % F 7 (8 G a 1 & #$ (' V(')( ! ' m 1 &- - - - " ¤ O'65_7X! % a 1 F 7(* G N
' & ( & 7 ( m ( ' & ( m (* !
m !" ¢ ')( $ ! % $ 5n ! % F G a 1 0& 3 (' V( (*' ' m !' - - #$
1 & O'65_7X! % 1A- ¢ 3h
N ¤ W 7 ( m (*' & ( m ( ' & W &
! -Q1 m !" !" £) ¢ ')( $ ! % $ 5n'65_7X! % ( m ! " ( ! " 0& ' ] ( m ( " ¤ '! % m ( ! ' ! ! ' ( ! £ & N (' 7 ( 1 ( (* m ' '
m ' #7( (*! ' &
' $L ' ')( ( $ $ 5_5 7 7 0& £ & ¤ &-
TJ m (87! " 7 43 ¢ 3 )£ ¢ $9% ')( $ 5_ X7 ! % C7 ( m ( '! " & - & m ! ' 0& ¤ 3 395
20.
21.
22.
23.
24.
25.
) ' * '
' 5_7 7 6 ( $ ! % (*'! " &- & $ L J (* ! % 3 £ S ') 'g5 7! % 3 3 ¢ SEa ¢ 3h 1 ( ' ( 43 ¢ £ 3 ¢ J $&% 1 L "
&- 0& 43 £ £ & ')( $9$ % 5_7 F]3 O'65: G 0& & -!&
S
& SE- a 1 S 0&
=
'I@ B
¢ j + -21 lM1 -; . ¢ 3 V ¤ . . . E(:7 ¤ G j + + -; & l + 1A-; & F
SEa & - 0& ST & & ¤ O'65n £ $ 5n L & - &- a 1 & ¢ X £ & 1A- & ¤ J O'65_7 ']" " (87 O '65_7 $ 5_7 L O ]! " &- 0& ' '#" &-21 0& = '8>?7A@CB J
5.8.5. Sums containing products of Un (z) 1.
2.
¢
X & 7 7(8 £ & 0 3 £ ¢ X £ 4 0& a ¢ & 5h' L " ¢ J
43 '8( $ ! % $ 5h' 5h ! % & ( ' L " 0& J ' ' % C7E(_ ! " ' 5h ! % ( 1 L' j 0& 3 J
5.8.6. Sums containing Un (ϕ(k, z)) 1.
2.
1 0 & ¤ a 1 0& j a 1 & 0 ¤
¢ ' & ¡ 'I@CB S $ L 4 3 = J S ¢ '$ L E & 43 £ & £ & a a 1 Q ¤ 4 3 S a S a J 3 3;
+
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
¢ ' 43 J $ L S ¢ ' $L S 4 3 J 1 ¢ ' 43 J $ L 1
; L1 :
¢ & 3 £ & S S l ¢ 3 ¢ S ¢ S 3 £ & S S F $ G l $ 5 3h& S S:F $ (8 G 43 ¢ S ¢ i S l 3 ¢ & S
¢ ' S / $ L 4 3
J ¢ ' 43 J $L -2 1
1
43 ¢ S _ £ & 4 S S l
¢ S _ £ & 4 SEa
SEa 1 / 3 3 £ 3 ¢ S ? ¢ ' S S F $ G S l 3 £ & 4 S $ L 4 3
J 1 ¢ ' $L SEa 1 SEa F G £ ¢ S 4 3
1 $ l J 1
= '8> = '8>
= '8>
@CB
= '8>
@ B
= '*>
¢ ' $ L S S #F " ¢ $ G 3 ¢ S ? £ ¢ S 3 V& S 3
l J 1 = '*> ¢ ' $ ( & ¢ $ G £ 3 ¢ S ¢ S E $ 4 3 L S a F " n 5 $
1 l J 1 £ S E a S 3 1 = '8> ¢ ' 5 , 43 ¢ SEa 1 ? S l $ $ 4 3 L & S S )+* J 1 3 £ ¢ S = '8> ¢ ' SEa n 5 , $ $ 4 3 L & 1 SEa 1 ) * J 1 £ 43 ¢ S ¢ / S 3 £ 43 ¢ S ¢ 4 SEa 1 = '*> l 3 3 :
@CB
= '8> 3 £ & SEa 1
@CB
= '8> 1 S l ¢ 4
@CB
@CB
@CB
@CB
@CB
@CB
@CB
14.
15.
16.
, '
$ ¢ ' 43 ¢ S ? £ S $ L 4 3 & S * $ 5 , ) J 1 3 $ ¢ ' & SEa 1 SEa 5 , $ $ L
4 3 2 1
1) * J 1 £ 3 ¢ S ¢ S 3 £ ¢ l " ( 7! O' ¢ f ¢ # 5 7 $ L $ f &- -21 ) J
¢ S l S 43& = '*>
@CB
X3& S = '8> @CB $ m 5_7 $ m 5_7X! 5 , ( ! ' O'65_7X!A ' m 5_7X! ¤
5.9. The Hermite Polynomials Hn (z) 5.9.1. Sums containing Hm 1.
2.
& -21
3.
4.
5.
6.
7.
"
nk (z)
2(:7X! ' ¢ ')( $ ! % $ ! % & O'! % 3 * 2(:7 , ¤ ) ')( $ ! ' '" £ ')( $ ! % $ ! % 0& & ')(* $ ! % £ & - ¤ ' " " ' % $J L m ! " & 43 ¢ m ! ' 1 - -; O ¤ ( ( #$ 1 (*' L " 1 ' L ; 1 ¤ ')J ( $ ! % $ ! % - & J ' % - N &
1 1 !&
m !" ')( $ ! % $ ! % 3P
- & (' #$ 1- - &- - ¤ m' ! % ' - N & & 7 ( m ( ' 1 ( m (*' 1 !
( ' ( ' ' ' -21 7 ' % ')( $ ! % $ ! % 0& O'! % N 1 V7 ¤ J$ L J $ L
! (' m ( m !' m !" ')( $ ! % $ ! % ! " 0& ' % ! ' N m ( (*'6! 5_7 1 ¤
3 3<
+
8.
9.
10.
11.
12.
13.
14.
15.
16.
¢ 43
(']")"'&
$ ! %
; P1
( ' 1 ( ' ] &B B B ! ' ¢ ¤ & 4 3 '! % ES a N S 7 & B B B #7 ( "
1 2- 1 !
" ')( $ ! % $ 5_7X! % a 1 0&
" '65_7! % 3 ¢ a 1 a 2(:7 ¤ ,
1 ) * '" & ')( $ ! '%& £ a ')( $ ! % $ 5_7X! % a 1 0& & 1 O')(* $ 5_7X! % £ & - ¤ m !" ¢ ')( $ ! % $ 5_7! % - a 1 & 4 3 (' £ m ' ! % ' 1A- N & & 7 (
&<- - ( ' m ( ' & ( m *
#$
- ¤ &
!
m !" ')( $ ! % $ 5_7X! % 3P
- a 1 & (' #$ <& - - - - ¤ £ m ' ! % ' 1A- N & & 7 ( m ( ' & ( m (*' & !
( ' V ( ' O ' 7 ' -21 ')( $ ! % $ 5_7X! % a 0& £ '% ! % N 7 ' ¤ $ L * $ L
1
& ! J J
" "" ' ' % ¢ ! ¤ ! J $L m " a 1 0& 43 m ' &4 - -; 0
( '<( 1 L)" J ')( $ ! % $ 5_7X! % - a 1 &
( 1;- ! ( £ J & ' L % ' 1A- N
& &
&
( ' ( m !' m !" m £ ')( $ ! % $ 5_7X! % ! " a 1 0& ' % ! ' N m ( (*'6! 5_7 3 3 L
&
#$
¤
¤
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
, '
¢ (#' ")"'& _ 5 7X! % a 1 0 & 43 $ (' ( 1 (*' ] &B B B ] ! ' & 43 ¢ O '6 5_7X! %]SEa 1 N S -21 7&B B B #7 ( " ! " " ' $ 5 ! % a ST & 43 ¢ SEa £ S SE-;&a -21
¤ $ L J ( (*' m M7( m ! ' ' m 5 !" ¢ S 5n ! 5n'! % N m (*' 1 ¤ J $ L $ % a S & 43
! m 5 !" ' $ 5n 5_7X! % a SEa 1 0& $ L J ( (*' m £ 3 ¢ S C7 ( 5nm '! '! % N
m ( ' &!
' " " ¢ SEa £ SEa 1 1 -;
¤ 5 ! SEa J $ L $ % a SEa 1 & 43 ¤ ¢ ' $ ! % ! ' ! &-21
) , 43 J $ L $ % & 1 L ' J
'%& ' % a ¤ ¢ ' $ ! % a 5_7X! % a 1 & 1 1 ) , 43 J $L $ & L ' J
" " (' 1 ( ' m ( 1 O ]! ' $9% ') ( $ ! % ! " " ¤
' ! % 0 & N : ( 7 (
m m
&- & 1 ! " " $9% ')( $ 5_7! % ! " m
&- a 1 & ( ' ( 1 (*' m ( 1 ! ' & O '6 5_7X! % & N 1 m (: 7 ( "
! " 7 . ' 5_7 F 3 G &- 0& j + - lM1 $ $ L J ( ' 1 (*' 5n £ & N ( " & 1 ! 3 3 P
¤
¤
¤
¤
+
27.
28.
7 . ' " 5 7 F]3 3 G &- a 1 & j + - lM1 $ $ L KJ ( ' & ('*( 1 5h £ & a 1 N ¤ ( " & 1 ! " 7 ')( $ ! % SEa &- & ' % S & & b # 0 ¤ Q $ L J
5.9.2. Sums containing Hm 1.
2.
3.
4.
5.
6.
; P1 ,
nk (z)
and special functions
'65_7 $ 5_7 L #& - &- a 1 & J 7 ' '# " " (8 & " $L ]!
& J
'65_7 '#" '#" 0& ¤
'
]! '#"
&-21 &
= '8>?7A@CB
7 7 $ ! % 'T( $ 5_7X! % 0 f63 0& O'65: O! % fg3 3 ¢ (' ( '65 1 V7#7 ! ' Y £ & a 3 "
a 0& O'! % m (*')(:7X! N m (*' ] ! ( 7 7 $ 5_7X! % ')( $ 5_7! % 0fg3 a 1 & O'65 ! % fg3 3 ¢ ( ' ( ' ( 1 V7#7 O ]! '%& Y £ & a & 3 "
6 ' _ 5 7 ! * ( ) ' : ( X 7 ! a % 0 & N m
&
m ( ' ] ! ( C7 ( m ! " ¢ C7 ( m ! " ¢ a ! O ' ! $ % 0fg3 & 43 1 % ')( m 5_7X! £ & 43 ( ' ( 'g5 m (*')(87 ( " 1 ( ' Y fT3 3 ¢ fg3 3 ¢ N ! m
& 1 (' V('65 1 m (*')(:7 m (*')(:7 N
m (* ' m (*' ! ( " ¢ C7 ( m ! " $ 5_7X! % 0fg3 a 1 & 4 3 43 ¢ a 1 '65_ C7X7 ! % ( ')m (! ' 5 7! £ & 4 a 1 m 395N
¤
¤
¤
, '
(' V(')( 1 m (*')(:7 m (*' ! ( " (' V(')( 1 m (*')(:7 m (*')(:7 N
¤ m (* ' m (*' ! ( " ( 7! ' ¢ ' ")" £ 5_7! " a / & & 5_7X! ' a / £ & ¤ 7. 43 J $L ' £ & A& 3 £ ¤ / G $ L F 8. - / & J " ( (:7 ' ( $ ! % $ ! % j &- Q & 9. W W # ( ' ( ' ( $ ! £ J 1 L ' 3 ¢ N ¤ W
' % ! -Q1 (1 *' 1
" ( (:7 ')( $ ! % $ 5_7X! % j &- Q a 1 & 10. W W # ( ' ( ' ( $ ! £ a 1 J ' 1 L ! ' 3 ¢ N ¤ W
% -Q1 (1 *' &
' 7 ( X 7 ! ')( $ ! % ( 5 7! c 3 ¢ ¤ 11. " &- 0& ( P5_7! ' - c -; 0& Y
0fg3 3 ¢ fg3 3 ¢ & N 1
5.9.3. Sums containing products of Hm
"
1.
2.
3.
4.
nk (z)
' J( 1L $ ! % & 0 3# -21 -; ¤ $ L J " ( 7X! ' '%& ' J( 1L
5 7! O'65 7 1 -;
¤ J $L $ % a 1 & ' $ 43 A J L &- A& -21 & 1 £ + &-21 . 43A W £ E3 W / -;&-21 ' ¤ J $ L Q &- & 43V
395)
& A& ¤
+
5.
6.
' J $ ' J $
7.
£
8.
£
9.
£
L & L &-
a Q
1 Q
; P1 5
¢ £ a ¤
a 1 0& 43 Q
a 1 & 43 ¢ £ a 1 1 O ¤ ' 1 (*' L " J$ ! % ')( $ ! % Q &- 0& ' %
) 1 (*' L" J 7X! % ')( $ ! % a Q $ 5_
1
&- & ' '& ' ( % 7X O! '6 5_7X!
a 1 a F
1 ( 1 (*' L " J $ 5_7X ! % ')( $ ! % a 1 Q &- a 1 & ' '%& ( 7X'6! 5_ 7X! %
a 1 a F
1 5
(
, ¤
5 G ¤
5 G ¤
5.9.4. Sums containing Hn (ϕ(k, z)) 1.
2.
3.
4.
¢ ' 43 J $ L ¢ ' 43 J $ L
= 'I@CB
ST & ¡ S /
" S 43 ¢ S £ i £ & 4 -; (*') ( ( $ ! % ! $ 5n '! % ¤ a
( 7! " ' / 4 3 ¢ SEa 1 £ SEa 1 F $ L E S a $
1 J 1 S ¢ £ ¢ i £ & a 1 -; 43 S a ¢ ' $ L SEa / 4 3
J S 43# SEa a 395,
G S
( ! " ( ' ( $ ! % $ 5n '65_7! % ¤
( ]! " $ 5:'! %
a ¤ S - & -21 Q
5.
, '
( X7 ! " $ 5 3 ¢
' J $ L ES a a SEa £ SEa a
1 / S ! " a a 1 $ ( n ¤ 5 ' ! % S 1 Q a -
£ S ¢ ' S $ 4 3 L S F $ G 3 & 4
J 1 S -; ¢ "'' &('
£ i £ S $ 5n'! % (* O')(* $ ! % £ & -
& 4 - 3 £ SEa ¢ ' SEa $ L E S a 4 3
F 7. 1
1 $ G 3Q & 1 J 1 S -; ¢ "'' &('
£ ¢ i £ S a $ 5n'! % ( O')(* $ 5_7X! % £ & -
& 4 - 1 3 ¢ '$ L 8. 3 f O SEa SEa F $ m 5 G J " ( ' % 5n '! % ( L" S 1 m f S a '65 J $ ! % ( $ ! % F G S - F G 1 ( J L ¢ ' $L f O SEa a 1 SEa a F 4 3 9. $ m 5 G
1 J " ( ' % 5: O'65_7X! % f SEa 1
& ( J L ( ( S L " 1 Y a F G a '6J 5 $ ! % ( $ ! % F m G S
- 1 6.
5.9.5. Sums containing Hm 1.
2.
nk (ϕ(k,
$ m ! " $ m ( ! ']")"" F $ G ')( $ ! % $ ! %
3953
¤
¤
¤
z))
' ]' " ' ( X7 ! ' % O'! % ' m ( ! F " m " $ "'& $ m ( ! ']")"" ')( $ ! % $ 5_7X! % a F
1 $ G ' '#" " '%& 1 # ( 7! ' % O'6m 5_7X! % ' ( ! a OF " m
1
1
¤
m ¤ G m ¤ G
+
3.
4.
5.
6.
7.
8.
$ ']" O')(* $ ! % &- F $ G $ & % 1 $ '#")" $ 5_7! "" O')( $ ! % &- F $ & $ % 1 $ '#")"'& $ 5_7X! "" O')(* $ 5_7X! % &- a $&% 1 $ '#" O ) ' * ( _ 5 X 7 ! $ % &- a &$ % 1 $ m 5_7X! '#" ')( $ ! % $ ! % F $ m 5 $ m 5 7! ']" ')( $ ! % $ 5_7X! % a 1 F
; P1 :
]! '#" O O') (* ! %
O ]! ' ( 7X! ' G 3 O'! % O'! % A& ¤ F1 $ G ( 7! ' ! ' & 3 O '6 5_7X! % 3 '65_7X! % a 1 A& ¤ ! ']" ' >?7A@CB ')(:7X! % =8 1F $ G O ]! ' 7 G '! % ' m 5 7! ¤ O ]! '%& O'65 7! % ' m 5_7! ¤ $ m 5_7 G
5.9.6. Sums containing products of Hm 1.
2.
3.
= '8>?7A@CB
¢ ' ¡ 43 J $ L& ST & ( 7! " $ 5_7X! '#" 5_7X! % O'T(* $ ! % &- F $ 5_7 $ '%& O'6 5n ! % a
( 7! " $ 5_7X! ' $ 5_7X! % O'T(* $ 5_7X! % &- a 1 F '%& '65 ! % D a 43 ¢
nk (ϕ(k,
z))
G
=
a F $ _ 5 7
1 a 3
1 $ 5_7 G a 1 F a
1
G a 1) $ 5_7 G
'I@CB
(8 5 , ¤
a F 5 GIH ¤
5.10. The Laguerre Polynomials Lλn (z) λ nk 5.10.1. Sums containing Lm
1.
(z)
( ! ' ('! " '! " 5 ! 5n' ! % S ;- & $&% $ % S 0& 395 5
= > 'I@CB
# . '/
2.
¢ ' a 0& 43 ¢ $ L 4 3 Sc J $ ' 5 _ 5 7! " 3P& Sc a $ L J % 5 7 5_7!
3.
4.
5.
6.
7.
8.
Sc a -; 0&
= > 'I@CB
0&
a ¤ Sc a & - 0& ' 3 3 E
1 ' ! " m ¢ (E5 : 5 7! " Sc a & $ L 3 J ( m 58' 5:7 ¤ (E5n7! (% (E ( 5n( 7Xm ! 5n7! ' N ( 5*' 5n 7 ( m 5n! 7 L '
J ( ]! " 5n'! % ' ¤ ( Q(*'65_7X! " Sc a & 43 ¢ % ( &! ' SE $ c L ; & a J 5n'! % ( ! " ' ¤ ( P5 5_7! " Sc a & % ( P5 5_7! ' SE $ L c 0 & a J ( 5 7! ( ( ( ' m 5n' ¤ ' m 5 5n'! " m ! " ( P5 5 7! " 43& Sc a & $ % L N
J 5_7 m ! L J ( g5 5h' 5 7X! " ' ( g5 7X! " ( 65 5 7! " 3 Sc a 0& $ L J ' % % ( g 5 5h7X! '! ( % g( g5 5 7X! 7X( ! ( c 3& c & ¤ ' & ' SEa
Sc - & ¢ ' 10. 43 J $L ¢ ' 11. 43 J $L ¢ ' 12. 43 J $L 9.
SEc a 1 0 & 3 Sc - &
¤ SEc -;a &1 -21 0&
Sc -; 0&
= > 'I@CB
' ( Q( ! " C7 (Q(*'! " Sc - 0 & ( 9! ' Sc a ;- 0 & = > 'I@CB ( 6( ! " C7 ( m ( '! " Sc - & ( 9! ' ( # 6( m 5_7 ¤ ( P5_7! ( % m ! % ( '6! 5_7 L m '
N J 5 7 # 6( m * 395;
.
13.
14.
15.
16.
' (' ( J $ L
!" m 5 m !"
58'! "
3.
4.
5.
6.
( ' ] ( ( m 58' ¤ ` m ! ( "
nk (z)
I5_7X! ' 4 ')( $ ! % (P5_7! " c 0& ( 5 7! ' c F I5_7 G ¤ ')( $ ! ' ' ¤ & -21 ¢ ')( $ ! % ( P5_7! " c 0& ' % (P5_7! ' 3 43 3 d)3 ')( $ ! % ( ' (' ¤ ¢ 5_7#! 7 L 43 = ')( $ ! % @ ( P5_7! " c & N J m !" ')( $ ! % ( P5_7X! " - c & ( ' _ 5 7 ¤ a ! ' m c - - 1 c - - ' % -; N ! & & 5_7V( m (*'65_7#6 ( m (*'65_7 ( m !' m !" (' m ¢ ) ' ( ! ! ( 5 7 ! ' % ! ' N m ( (*'65_! 7# 5_7 L ¤ $ % " 4 3 c 0& " J '( 6( '! " ')( $ ! % ( P5_7! " 3P& - c & #$ ( 7X! ' 1;- ! ( ( P5_ ' % -; N & a 5_7 # P5_7 ¤ c 1 c
395: "
2.
¢ ' ('6( ! " m ! " Sc - & $ 4 3 L J ( 6( m ( ¤ (P5 m 5 % ! ! ' ( ]! N ( 7 (%V( 6Q( ( ( #7 ( ( m ' m ' ! ( " ` & 1 ^ 5n'! % ' - Sc - 0& % 3P& -; SEc - a 0& ¤ $ L 3 2 d J " ( ]! ' ( 5 5_7X! " $ 5 7! Sc a 0& $ L J " 7X! % 'g 5 7 D d c -21 0& 3 % ( P5n5 '65_5_ 7X! ' SEc -2a 1 a 0 & H ¤ S 1
5.10.2. Sums containing Lλ m 1.
Sc - & ( ( ]! % N ( & 1 ^
-
; N1 ,
# . '/
(']")"'&
( ' ( ' (% &BB B ¤ ( ]! ' (P5_7X! " c & ' % ( P5_7X! ' ES a 1 N S 2- 1 7 &B B B V7 ( " 7. ` ^ ! ( 6( '! " ) ' ( $ ! % ( P5_7X! " 3P& - c & 8. #$ ( 7X! ' 1;- ! ( (P5_ ' % -; N & a 5_7 #P5_7 ¤ c 1 c
' $ ! % ' O ] ! % ¢ ' ( P5_7! " c 0& ( P5_7X! ' c a F ,G ¤ 9.
43 J $ L ¢ ' 3 ¢ c -; & ¤ $ L 10. 4 3 c a2S & SEa J ' $ 5 !% % ¢ ' ( P5 5_7! " c a2S 0& ( 5 5_7X! ' Sc a 0& ¤ 11. 43 J $L m 5 !" ¢ ' ( P5 5_7! " c a2S 0& $ L 12. 4 3 J 7X! ( ( (*' m ¤ C7 ( m ! ' 5n( P'5_ ! % N J 5 7 m (*! ' L S ( Q(*'! " 5 ! 13. a2S $ % &c - 0& ( & ' "( S ( P5n'65_7X! ( 43 ¢ C$&7 % ( $ !( $ ! % c ¤ 7 ( G $ F E S a " ( ! ')( $ ! % $ ! % 3Pd 3 Sc - & 14. #$ ( ' ( ( 6( ]'65 1 ( ]% ' ! % N " = > 'I@CB
( & 1 1 "
! ( ! ) ' ( ! % $ 5_7X! % 3 d 3 Sc - 0& $ 15. ( ' #$ ( ( 6( ]'65 & ( ]% ' ! % N " = > 'I@CB
& 1 & ! (
¢ ' - S -21 0& S- S -;&-21 0& = > 'I@CB 16. 43 J $ L S
395<
.
17.
; N1 3
" '#" J 1&$ L % " & 43 ¢ ' % F#" G a F" G ¤ & -
1
nk 5.10.3. Sums containing Lλ m pk (z)
1.
2.
3.
4.
5.
6.
7.
( 7X! ' ¢ m 5 $ ! ']" a ')( $ ! % m 5 $ ! " c & m n 5 ' 65n')(:7 c - -; a 1 0& ¤ 43 ( ! " ' % ')( $ ! % ( P5_7! " 65_7! " c a a 0& ( P5_7! ' 65_7X! ' c 0& & ¤ m !" ¢ ) ' ( ! ( 5 7! " c a 0& $ % 4 3 ( O' m (Q(*' ' % ( m 5 ! ' 7! N (' V( 76 ( ` ' & 1 ^ m (*' ! " ' J '65 1 L " ( 5 7 ! " 43V& c a 0& $ L J '65 ( O' ' J % ( P5_ 1 7XL ! ' £ & N ( O' ( V( ' ' ((' 6 " ` ' & 1 ^ ! m !" a 3 # & & c ) ' ( ! ( g ' 5 ( P _ 5 X 7 ! $ % " m 1L" J
( O' 1 ( m (*' V'( 6(* ' m !' 7 (* m (* O' ( " ' % 1 ( m L ' ( P5_7! ' 3#
& N 1 ! J
1 ( m L)" J ')( $ ! % C7( m (*'! " ( P5_7X! " 3#& c a & ( ' % J ! 1 ( Pm 5_L ' 7X! 3V
N ( ' m ( ( O' ' ('( 6" (* ' ' ` m ' m & 1 ^ ! m !" a 3 # & & c ) ' ( ! ( g ' 5 ( P _ 5 X 7 ! $ % " m 1L" J
( ' ] m '7 % N !
m (*'65 1 P5_7
395L
¤
¤
¤
¤
¤
8.
# . '/
5 '65 1 L " ' J n ( 5 5_ 7X! " #& c a2a S 0 & $ & $ % L J (P5_7% ! (
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
' a 0& $ L Sc J -
N J
(
(* O' 5n O'65_7 ¤ 5_7 #P 5_7 L !
= > 'I@CB
Sc a &
#$ ( '65 ( ]! " & ¤ (P5_7! ( a ')( $ ! % $ 5_7X! % Sc 0&
%' % N
& # 5 7 !
' 7 5n Sc a 0& $ $ L J '65_7! 7 '65n O! c - & ¢ c a 0& 3 c a &-21 0& = > 'I@CB SEa
SEa 1 SEa
" ( 7! " S S S a ( $ ! % SEc - a S 0Q3 ¤ 9 $ % a2S $ 5 ! % c & 3 & " ' a > ' > @CB ')( $ ! % Sc a & $ L 0 & = Sc ! J 1 ' a & 43 ¢ 3 43Pd 3 Sc & $ L 3 Sc -; J = > 'I@ B ( ]! " ( 5 7! ( ' > 'I@CB ( 6( 'g5 7! " Sc a & ( 6( 'g5 7! ( Sc -; 0& $ L = J " '65 1 L ( ( ]! a ¢ S J ')( $ ! % $ ! % S -21 0& 43 ! % ' % ST0/ ¤ $ 5 ( 5 5_7X! " 3& Sc a 0& '%& 1 3 c a 1 0& 3 (P 5 ( ! 5_7! c a a 1 & = _'8>?7A@ B ' S -;&-21 S -21 ( '65 #$ ( ]! " ( P5_7X! ( & a > 'I@CB ')( $ ! % $ 5_7X! % Sc & ' % N P5_7 % = & !
395P
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
' J $ L
.
)" " ( Q(* ! " a (1 Q (* (*' c S - &
( L " ( V( 6( V( 6( ! % N " ( Q ( (* ' ( & 1 ^ ! " 7 ')( $ ! % c - 0& c -; F Q3 G ¤ (' V( m ! ' ( &! ' ' % -; N 7( m (*' 0fI 4 3& - c - 0& J
J
4 3Pd 3 S&- a ')( $
; N1 3
(*' `
= > 'I@ B
m ( ' ! Q(*'65_7 L ¤
¢ 3P& - c - & 43Pd2 43& -; c -; a 1 & ¤ ( &! ' (' &B BB 1 3P& - c - 0& ' % -; S N S J Q(*'65_7 #7&B B ! B #7 L ¤ 7 ' ¤ ! % C7 ( Q(*'! " c - 0& ' % ( 9! '
( 1;- V( ( 7X! " ( ! ' ')( $ ! % (Q(*'65_7X! " c - 0& ' % ( &! ' & N ¤
" ' ¤ ('! " & -21 ')')( ( $ ! ! ( ' &! ' ; $ % % P 3 & (Q(*'65_7X! " 43& c - & ( m !' m !" (' m V( ( " ¤ ')( $ ! % ! " - c - & ' % ! ' & N 1 m ( (*'6! 5_7 ` ^ " ) " ')( $ ! % m ( $ ! c - 0& (' #7 ' % ( &(*! ' '! 3P& -; N * ( 6 ' 5_7 #Q! (*'65_7 L ¤ m m
J )" " ' % ')( $ ! % $ ( m ! & - & ( m ! ' & 3P& -; -; 0& ¤ m 5 1 L' '( 9! ' ( ! "' 1 ( m (*' L" J ' % ( J
m ')( $ ! % & (*' L " - c - & 1L' J
J
(' m (' m Y £ fg3 ¢ N ! ! 'g5 7 ¤ N # 6 * ( 6 ' _ 5 7 5 # 6 ( ( m m
1 1
3;9N
# . '/
( Q ( 1 L" J
31. c - & * ( ' ) ' ( ! $ % & " L J ( &! ' ( ]! "' £ ¢ ¤ a #( Q(*'65_7X! ( 1 L ' d c -;&-21 & c -; 1 & J
' $ L 0& SEc a 0& ¤ 32. c a2 S J ' - c a2- S & $ L I f 33. J ( ( ¤ ( m 5_5n7! '' ! % ( ]! N ( (*( ' m ( ( ' V((' 6 " ` m & 1 ^ ! ( P5_7X! ( ' ( 9! ' " ' ' (6(*'65_7X! ( SEc - a 0& ¤ $ L 3 2 d & 34. c a2 S J ' ( Q( ! " $ # & L 35. Sc - & ( * ( ' J 1 L " ( J
( ( (*' V'( 6( ( ! ( (* O' ( " % N ` = > '8>?7A@CB & 1 ^ ! ' 5 & A ! ( 9 ! ( &! ( & $ " a ( 5 _ 5 X 7 ! ( P n 5 6 ' _ 5 7 ! ( P n 5 '65_7! ( Sc a &
36. " " Sc - 0& J $L = > 'I@ B " ( 5 7! ( ( 5 5_7X! " Sc a 0& 37. % ( #$ ' ( #$ c 5_7'g5 c 5_7 ( ! ' & ( ! '%& ! ! Y N ( P5_7X! ' & N '65 5: '65 5n
c 5_7 3 c
= > 'I@B W " ( ' J ' L 5 5_7! Sc a & $ L 38. ( * ( " J 1 L" J
( P _ 5 7 ! ( V( (*' ( > 'I@CB ! % N
J ( (* ' # 5_7 L = " ( ' m ( J L ( 5 7! ' a ! 7 ¤ ! ' % N m (:7# 1 5_ 39. $9% m " &c -
& 3;
.
40.
41.
42.
43.
44.
45.
46.
47.
48.
; N1 3
(' c * ( ' #$
7 # 6! ( O' ¤ ( g ' 5 c
7 F]3 G c - & F G F G
c - R a 1 & ¤ ('9! " ( O'! " F]3 G c - & ' m ¤ m ' ! % ' '( ! 9! ' - N (# 6 ( O'6! 5_7 L m ' m
J ( 9! " 'g( $ ! % m ! " - c - & #$ (' a c - a c - a 1 ! ¤ ' ( % 9! ! ' 3P& -; N m ' & & c a 1 ( ' c ( ' 5n7 m 5 ( '
' ( &! " 'Q( $ ! % c - & ( 5 4 , 5 5 ¤ 3 c - , c - ) , ) '( 9! " )" " ( &! '%& £) ')( $ ! % ')(P5_7X! " c - & ' % 43& -; ¤ 3 2 d J m 5n')( 1 L)" ( &! " G c - & ')( $ ! % m ! " ] F 3 ' 7 % N ( O' O'65n( m " (:7V( ` ¤ m! & 1 ^ O')( $ 5 m !A '( 9! " ')( $ ! % C7 ( m ( '! " - c - 0& " ' (' c - (*' c - a 1 (*' #$ '( 9' ! % ' m !
( m (* O! '65_7 ¤ m ' & N & c a 1 (*' c (*'65_7# 6
C7 ( m (*'! " ( &! " £) % ( 5n'! " 43& - c - & 3 d2 m ( 9! ' (' m (:7 & m % m ! ' '( % & ! ' 3P& -; N J m # Q(* O' ! L ¤ m ( &! ' 3P& - 4 P 3 2 d ( Q( 1 L" J ') ')( $ ! % J1 ( m ( ' L" ') ( $ ! % C7 ( m
('9! ' c - 0& ' % ;- N
3; ,
49.
50.
51.
52.
53.
54.
55.
56.
# . '/
$ ( ( 9! " ')( $ ! % ')(P5_7X! " - c (6(* $ ! : ( 7 &! '%& ' (Q ( * ( O'! (:7 3P& -; %
& ( ' #7 N c a 1 ( ' c ! a & (*' ¤
1 ( m L " ('9! " J ')( $ ! % C7 ( m (*'! " F]3 G c - & ( ('9! J 1 ' m % L ' ! ' F]3 G N ((* O ' ' m Q(*(*' ! '6 5_7 L ¤ m '
J m ( &! " £ - c - & ) ' ( ! ! ( % ( 6 * ( ' $ % m " & m L" J
(' m 5 5:' (87 ( m (%6(*' C ( & ! ' ¤ 6( 'g5 7 ( 1 ' % m ! ' m 5 6( 1 L ' F G & N 1 ! J
C ( & ! " £ - c - & ) ' ( ! C 7 ( * ( ' ! ( P 5 $ % m m " 1 L " J
( & ! (' ] m ( P5n' #6( m ( '65_7 ¤ ' G Q (*'65_7 ( " F ` ' % m ! ' m (P5 1 L ' &N 1 ^ ! J
m ! " ! " ( &! " £! ')( $ ! % C7 ( m % ( 9! " C7 ( ( &! " ') ( P5_7X! " - c - & 3 2 d ! ' '( 9! ' & ( ' V7 ( m ( % ( ¤ ' % C7 ( m ! ' ( & ! C 7 ( ( & ! 7 ( ( ' V 7 ( (*! ' L P 3 & N ; ' ' m m
J W " ( ' J ' L 5 5_7! Sc a & $ L ( * ( " J 1 L" J
( 5 7 ! ( V( ( ' ( > '8>?7A@CB ! %
N J ( (* O' # P5_7 L = ' 7 $ L F G 3 d 3 F]3 G c a2- S & J ( ( ]! % N ( ( O' 5 5n7 '6( 5_" 7 '( 6( ` ¤ & 1 ^ ! 7 7 $ @& !C@CB ')( $ ! % (P5_7X! " c - 0& (P5_7! ' c &
$ L = = J
3;93
.
57.
C7 ( m (*'! " ')( $ ! % (P5_7! " - c - &
; N1 3
V7 ( m (* ' ¤ m ! ' O '( ! % ! ' ;- N ( Q' (* '65_7 ( L
Jm !
58.
59.
5n'65 1 L" ' JP ( 5 7! " £ c - 0& $ L J
P5 1 J 5_ 7XL ! ' ' C7 (* m (*'! " 1 L " ')( $ ! % C7 ( m (*'! J " 5_7X! "
60.
61.
62.
63.
64.
( ' ('6( O' V( 6(* ' ¤ ] ! ' O'! % & N 1 ( 6( ' ( " ` ^ ! c & W #$ ( ' 1 ( m ( ' ! ( & ! ' ' m ' % m ! ' 43V& -; N & m 5 a ! 5_7 ¤ 1 c - 1 c -
Q( $ 1 L " ( c L " '65 1;- c L " ')( $ ! % J (*J ' ')J ( 5_ 7 F G c - & c L" 1 J L" J
' % ( £ 3 ¢ a 1 J c L '%& F G D + c -21 . - F ,G9H ¤ 1 J L' O' ' ( $
£ R&-21 £ - - &-21 F G 3 7 ' ' L ¤
J O'65_7 £) ¢ ')( $ L - - a - 0&
1 J £ R a 1 £ ¢ - - &!- &4 F G ¤
a 1 ( Q(* ! " ' a $J L (1 Q(* (*' L " - c S - 0& ( J
( V ( 6 ( V( 6( (*' ! % N " ( Q ( ` = > 'I@ B (* ' ( & 1 ^ ! 5n')(* $ ! % ( $ ! % ')( $ ! % ( P5_7X! " SEc a a & &-
( P5_7X! ( & ' ( P5_7X! ( P5_7X! c 0& c & = > 'I@CB ( ' S 3;!5
# . '/
65.
m !" 2d 5 7 L " ( &$ % c ( m _ J
66.
c a J 5n L '6" 5_7X! (9! " "'& &c - & ( ' a ( #$ c 1 m! ( 5 ' % 7! ' N
c a 1 Q (* m 5_7
) ' * ( ! ( ' ( $ % 1 " L 43 ¢ £ 3 ¢ J%
- &-21 & & $ &-
£ a 1 F 7 G -;&-21 F G a 1 (' c & #$ ( &! '%& cL" & 0d ) $&% ( 5n ¤ ! a J & '65_7X! " & c & ' % N
c c & a 1 &!- &
C 7 ( * ( ' ! n 5 O ) ' ( " m m 1 " L ')( $ ! J % ( 5 7! " F G c - & & ( ' ( #7( m ! ' m 5: O')( 1 L ' m ( ' V'( 6(* ' #$ m 1 £ 'J ! % ( P5_7X! ' & & N
m (:7 ] (* m ( O' ( ! ! ' ( 5 " m m 1L" ')( J $ ! % ( P5_7 ! " F]3 G c - & & m ! ' ')( m 5 1 L ' ( &! ' J '! % F3 G ( ' m ( O' ( m (*' Y N ( ' V7 (* (*1 ' Q(*! '65_7 m
& & m ¤
¤
67.
68.
69.
¤
¤
nk 5.10.4. Sums containing Lλ m pk (z) and special functions
1.
2.
¢ ' ¢ 43 J $ L c 0& 3 ¢ ¢ c -; & & -21 ')( 7 $ c -; & ¤ C7 ( m ! ' C7 ( m ! " ( P5_7X! " fg3 c 0& ' % ( P5_7X! ' ')( m 5_7! 3P& (' m (*')(:7 V('6(*' Y 3hf- ¢ fg3 3 ¢ N ` m (*' ! ( " & 1 ^ (' m (*')(:7 m ( ')(:7 V('6(*' ¤ 3 N ^ ` m ( ' m (*' ! ( " 3; ;
.
; N1 5
( 7! " ')( $ 5_7 ! % (P5 7 ! " 0fg3 c & 3. '& ( 5 7 7 ! fg3 3 ¢ '6 5_7! % 3 ¢ c & '& a 1 ' (' ('6(*' #7V7 ( ! 1 ¤ ' % ( 5_7X! ' m (*')(:7X! N
m (*' )" " ('9! ' ) ' ( ! ' % 3P& -; fg3 $ % 0f63 c - & 4. ( &! '#" 7 (*' V7#7 3 3 ¢ ')(:7X! % m ( '! 1A-; & N & J m (*'65_7 #Q(*'6! 5n ] L = >_'I@CB ( &! ' )" " ')( $ ! % 0f63 £! c - 0& ' % 43& -; 5. 7( ' V7#7 '2 Y 0fg3 £ m (* '!A (Q(*'65_7! N ! 5n * ( 6 ' 5_7Q(*'6 & &
7 (*' V7#7 '2 * ( O 6 ' _ 5 X 7 A ! ( Q * ( 6 ' _ 5 X 7 ! m N& & a & (*' # 6(*'6! 5n ¤
" ' m ( O' ( ( 7! " ')( $ ! % 0fg3 1 & & - 3P& 3 ¢ m (*' 1 N 1 m ( '6! 5_7 L ¤ 6. J ( ! ' ¢ ( 7! " a ' % 3 d2 d 3 _ & ¤ & $ % d & P 3 & 7. c -; & C7% ( 9! ' ( 7! " ')( $ ! % d 3 1 & c - 3P& O' % 0d 3 _ & ¤ 8. " / £ c 0& 9. ')( $ ! % Q(* O'65 & L " c - a a 1
J
( 7X! ' £ ¤ ' % ( Q( 1 L ' c a a 1 / J
" " 5 8 ( 7 j Q c a 0& 9 $ % 10. & " ' (' ( ' '( 6(*' #$ ' ! '! % (87 N ¤ W % : ( 7 ( * ( ' , & 1 ) 1 ! - W -Q1
-Q1 3;9:
11.
12.
13.
14.
15.
16.
17.
# . '/
$ 5_7 7 ( $ 5n6 ' 5_7X! % j Q a 1 & ' % F G ¤ & ( ' 1 $ 5_7 7 a
5n'65_7X! % j Q 0 1 & ' % N 7#7 ( ¤ $ &! ' $ 5_7 ( 7X! $ 5n'65_7X! % a 1 Q a 1 0& '! % F 63 G ¤ & $ 5_7 ( 7X! ' a $ 5n'65_7X! % 43& Q 1 0& O'! % / ¤ & 5 ! ' a ¢ '( 6(*'! " 5 G ¤
c 1 $ ! % 4 3 Q F c & P 5
& 1L ' J
¢ '( 6(*'! " $ 5 7! % a 1 Q &c - & 43 ' & 5 ]! c a a 1 F 5 G ¤ 5 1
1 L '%& J
( P5_7! ' (' 5n O ¤ . ( ! " $ 5_7X! % j + - lM1 c a 0& ' % N
# 5_7 ! L J &
nk 5.10.5. Sums containing products of Lλ m pk (z)
1.
2.
( 7! " ( 7! ' O' @ @CB ')( $ ! % (P5_7! " c 0& ( 5_7X! ' ' L c -; 0& ==
J ( 7! ' " ' ¢ $&% " " O')( $ ! % &- -21 0& ' (*' -;&-21 #& ¤ 4 3 1 J L ¢ $&% ')( $ ! % - c - & - 0& 4 3 (' a (*' a (*'65_7 ' ( 9 ! ( ! ' ' c c 43 ¢ - ¤ ! ' % & N & Q(*'65_7 )(*'65_7 P5)(* '65_7 $&% ')( $ ! % - c - 0& c - 43& #$ ' ( 1;- ( ( 9 ! ' % - N ¤ !
& c - a 1 c - 5 7 # 6( 'g5 7
3.
4.
3; <
.
5.
('9! " !£ ')( $ &$ ! % % ') ( 5_7X! " 43 Q& d 3 3 ('9! " ! £ ')( $ ! % ') ( 5_7X! " - c d 3
; N1 ;
c - Q c - &
43Pd2 a 1 Q& -; c - & ¤
0&
6. '( 9! ' & 1 L ' 3 ' % J F G c - & ¤
' ( ( ' ! ( ] ! " . ( 5 7! " c Q & ( 5_7X! ' j + c lC- c - - &-21 F ¢ G ¤ 7. &( ' '( V( )(*' ¤ ( ! ' " " ' % &N 4 3 : 3 4 3 Q 0 & L 8. c - Q J & " (' ¤ 65_7! ' a ( P _ 5 X 7 ! ' P _ 5 7 ! 65_7 L % 9. " c Q &- 0& 1N J ")" ' ( $ ! % Sc - 43& - 0& 10. ( ()'P5*7% '! ( % ( ! ' 3P& -; N (T( 'P 58 P7 58Q7 ()! ( 'P 5*7 L ¤
J " ( Q ( ! ( ] ! ' a & - S -21 3P& ' % 9 $ % = > 'I@CB 11. Sc Sc -; 0& & ( 7X! ' ( ]! " a a ' % 43Pd 3 Sc -; & = > 'I@CB 9 $ % 12. & P 3 & Sc c -; & ( 7! " c - 0& SEa 0& 5:' L c & ¤ 9 $ % 3 d 3 13. S SEa J & ( ! ' a ( 7! " ' % Sc & ¤ 14. $9% 3 d 3 Sc - 0& &-;- &- c 0& ( ]! " a 0& a 0& 9 $ % 15. Sc &7 65n'65_7 ( ( P5_7! ( % ' 6% 5_7X! ' N P5 5_ 5 7 65_7 ! L = > 'I@B
J 5n' ( ]! " a 0& c a2SE a & 9 $ % = > 'I@CB 16. Sc J L SEc a & &3;9L
# . '/
17.
J
18.
19.
20.
21.
22.
23.
( ]! ' a > 'I@CB ' % Sc & $ L SEc - a 0& -;&- c 4 3& = -; & ( ]! " a 5n' > 'I@CB $9% Sc 0& c -; a & J L SEc -;a 0& = & ( 6( ! ' 5 $ S > 'I@CB ' % Sc -; & = J $ L SEc - a 0& &- - - c -21 43& $9% (Q(* $ ! '( 9! " ')( $ ! % ') ( 5 7! " 43 Q& - c - Q c - & 3 43Pd2 a Q& -; c - & ¤ 1 ( & ! " £) d2 ( P5n'65_7X! " F]3 G &c a - Q - c - & ( 9!' ' % & F ¢ G ¤ m 5n'! " ( &! " £ ) 3 d2 ')( $ ! % $ $&! %% ! C7 ( P5n'! " C7 ( m ( Q(*'! " Y - c - 0& c - 43& (' '65 V( m c 1; - c 1 ( #$ ¤ O ' ! m % ! ' ! ( &! 5 '%& 9! N '
m ' m m 5 1 V( ! "
$&% ( 9! " £ ) d2 ( P5n '65_7X! " F 3 G - c - 3 Q - c - Q c a & & ( ' c c a 1 ( 9!' ' % & N " ¤ & 1 !
5 $
nk 5.10.6. Sums containing Lλ m pk (ϕ(k, z))
1.
2.
3.
¢ ' 43 J $ L Sc ¢ ' 43 J $ L c & ¢ ' 43 J $ L Sc &
= 'I@CB
& ¡ ¤
S ( ! " ¤ ' % % ( ( P5 5_77! ! ( -; 5 ' L ( 5:g : ' 5 7 ! " ' a J $ 3;9P
.
¢ 4. 4 3 1 ¢ 43 S
' J $ L S Sc F ' % ( P5_7X! ( % ( P5_7X! ( "'
( ¢ E S a $ G 3 1 % S -; 5 ' S -; a J $ : S & a 7 G c - & ' m 5
; N1 :
L 3 3 d2 -
= > 'I@CB
( ]! " ' ¢ a a Q ¤ n 5 ' ! $ $ % L 4 3 5. c SE Sc a J m $ m 5 ! ']")"" m ')( $ ! % F G c -; F3 m G ¤ 6. F $ "" a ( 5: O! ']" ')(:7X! % = '8>?7A@ B 7. $&% &c - & 1 ( &! ' ( ]! "' ¤ $ m 5_7X! '#")"" ¢ ')( $ ! % - c - f & ' % ' m 5_7X! 8. 7 ( ! ' 7 $ " $ 5n'! % &-
& 3 # ' % ! -;&-21 F G ¤ 9. 1 7 7 $ "'& ( ]! ' $ 5n'! % & D -;&- F G 3 -;&-!& F G9H = '8> X@CB 10. & &-
&-21 1 "" $ $ 5n'! % &- & # ')(:7X! % ¤ 11. 1 $ "" n 5 ' ! ) ' * ( O ! # ) ' (:7X! % ¤ $ % % 12. &-
& 3 1 $ "" $ 5n'! % &-
& ] ')( ! % 3 ] ')(* ! % # 'T(:7! % ¤ 13. 1 " ( ' V7 ( m m" $ " $ 5n'! % &-
& # ' % ! N 7( m !V7<5 m ` 3 ¢ ¤ 14. 5 $ m ^ 1 7 $ 5 7! "" a £! ¢ $ 5:'g5 7! % 1
& ' % ¤ 15. & " " 7 ¤ $ 5 7! a £! ¢ : 5 g ' 5 7 ! ' # ) ' : ( 7! % $ %
% &- 1
& 3 16. 3 : N
# . '/
17.
18.
$ 5 7! "" 5:'g5 7! % $ $ 5_7X! "'& $ 5 7! 5 m $
19.
20.
21.
22.
23.
1
1
a 1 £! & " 5n'65_7X! %
7 ¤ ¢
& ' 7 % 3 7 ') (: 7X! % ] ')(* ! %
&- a 1 4 )£ ¢ 4
& ' C7,7 5 ! N ( ' V 7 ! ( a m % m &- &
$ 5 9! "" (9! " a £! ( 9 ! ' & $9% (P5n'65_7! " c d2 4
& ' % m ¤ & ( $ m ! " $ m 5 ! '#")"" ']" ')( $ ! % (P5_7X! " c F $ G 3 ' % ' m ( P5_7X! ' ' m 5 ! c F3 '$ ]" 5n' $&% 3 d 3 &c - F $ G 43 ¢ ')(:7X! % &-21 ¤ ' ( 7X! " ¢ ')( $ ! % ( P5_7X! " f &-21 c F $ m 5 7G ' % ' m 5_7X!A( P5_7! ' ¤
¢ ' $ L f O SEa c 3 SEa J d ¢ 43fI S
¤
G ¤
F $ m 5 G ¤ m S ( Q( ! " : 5 ' ! $ % G Sc F G a F 3
nk 5.10.7. Sums containing Lλ m pk (ϕ(k, z)) and special functions
1.
2.
$ 5_7! '#" ')(* $ ! % (P5_7X! " & '%& " '65_7X! % 1 L ' & J
] ' " $ 5 7! ')(* $ 5_7X! % (P5_7! " '& " '65 7! % L ' J
F $ ' 3
G 5 5_7 c F $ 7G & I " 5 ! %' & Q(:7X! ' & c -2a 1 F 5 G ¤
a F $ 5_7 G c F $ 5_7 G
& - 1 '%& 9 " 5 ]! ' & c -2a 1 F 5 G ¤ Q(:7! '%& 3
& 3 :1
.
3.
4.
5.
6.
$ 5_7! " " $ ! % F $ 5_7 G
; N1 L
c a 3 & & " $ ! % $ 5_7! c a ¢ 4& ¤ &-
$ 5_7! "" $ 5_7! % a 1 F $ 5 7 G
c a 4 3 4& & " $ 5n ! % c a ¢ 4& ¤ &-
7X! " " & -21 $ 5_ $ ! % 3 &- 4 3Pd 3 F $ 5_7 G c F ')( $ G & ( _ 5 X 7 ! ' 3 ¢ &-21 ¢ &-21 '! % F '65_7 G ('6(*'! " ¢ g ' 5 7 $ ! % $ 5_7X! &c - F , ) "" & -21 $ 5_7X! $ 5_7X! % 3 &- 3 d 3 a 1 F $ 5_7 G &c - F ')( 43 ¢ &-21 ¢ &-21 ( OP'Q5_5_77! '! % a F 'g5 7 G 1 ( Q(*'! " ¢ 6 ' _ 5 7 $ 5n ! % &c - F T , )
'65_7G ¤ $ G '65_7 G ¤
nk 5.10.8. Sums containing products of Lλ m pk (ϕ(k, z))
1.
2.
3.
¢ ' 43 J $ L
=
¡ Sc &
( ¢ ' / Sc 3 / % S l 43 J $L Sc ¢ ' S 4 3 J $L 1
Sc F $ G
'I@CB
= '8>
(P5_7X! ( ¢ S S l Sc ]F 3 $ G 43 % ( ¢ S 3 3 % !
@CB
3 :,
= '8>
@CB
0 ' /
4.
5.
6.
¢ ¢ &- -21 3:3 43 Q - c - 4 Q F $ 5_7 G &'%& ( ' : ( 7 V ( Q : ( 7 V( (*')(:7 ( ! '6_ " " 5 7X! % (P5_7!A 65:'g5 7! D & N L J '( 6(*'! " ¢ &-21 65_7X! " c F $ 5 7 G F $ 5_7 G & ' '& " ' ( 7 ! '65 7! % ( P5n'65_7X! 65 7! ( 5:'g5 7!Ij + c lC- c - - &- . F ¢ ' a 1 $ 5 7! "" ( g5 7X! " 43& c F $ 5 7 G a 4 3 & &!" $ 5 7 ! % ( 6 5 7! " a &-
3 ¢H ¤
G ¤
¢ 4& ¤
"
7.
J L 5_7 c - ¢ Q a 4 3 4 & $ & $ ( 5_9! 7X" ! % F3 G a 4 ¢ 4& ¤ & -
5.11. The Gegenbauer Polynomials Cnλ (z) λ nk
5.11.1. Sums containing Cm 1.
2.
3.
(z)
('! " '! " S 5 ! $&% $ % S - - 0 & ')( S '65 43 ¢ S ( % ! ' ! ' % £ & S - N '6 5_7
1 ! m !" ¢ ' C7 ( 9! " S c - 0& $ L 4 3 J ( (QS S V7 (Q( m ( 5n' 7% ( 6( m ( ! ' 1 ! ( 9% ! ( C7 C(% 6( ! ' ( Q ( 5n' #7(Q( m (
7 &N
S "1; - ¤
!
" ¤
(9! " (&! ( (P5 ! " c S a 0 & 4 3 ¢ SEa 1 -
1 ( J L£ . a ¢ Y D j SE+ -!a &4 l c &-21 3
<3 j ES + -!a &4 l c -!&4 . ¢ 3 £
H ¤ 1 1 3 : 3
.
4.
5.
6.
7.
8.
9.
¢ ' ( 9! " a ! 43 J $ L m " c S & ( Q( 43 ¢ S (&! ( m % m !'
!'
;
&N
(
P5 # Q( m 5 6( m 5 (*'65_7 1
!
5_7 ¤
( &! " ¢ ' ( P5 5 7! " c SEa a 0& $ L 4 3
1 J # ( & ! ( & . 3 ¢ a 1 j S + c -;a &-21 l a 1 £ 3 ¢ = > 'I@ B & ( J L ( 9! " ( P5 5_7! " c SEa a &
1 &! ( & P5 -21 D (P (5n '! ( & c SEa a & <3 6(87 c SE-2a 1 & H ¤ ¢ ' ( 9! " a ! 43 J $L m " c SEa 1 & ( # P5 5_7#6( m 5 5: ( Q( (:7! ' 3 ¢ S £ ( 9! ( & m ¤ 6( m 5 (*'65n & % m !' &N
! ( &! " ¢ ' c SEa a 0& $ 4 3 L P 5 5 J 1 L " 1 J
( ( P5_7! ( . £ 1A- S d2 J 1 L ' j S + c a &-21 lM1 -; £ 3 ¢ ¤ 5 & ( 1L( &' J L J
( 6( L " ' J1 ¢ 3 d2 3 ¢ - c - 0&
$ L
SEa 1 J ( & ( &! ( & 5_7X! % ('6( ! '
Y ; ¢ 3h O S 3 3 F 1 Y j SE+ - a c a - S -21Rl -; & 3 :95
7 3 d 3 G ¢ h 3 O a 1 . 5(877 5n'8> @CB , = )
10.
' J $ L
0 ' /
(9! " &! ( & ¢ c a &
3 E S a P5 5 1 " 5_7! % 5 1 L ( & '
1 L J J
¢ ¢ Y 0d 3h
3 4 3 1 5 '*> @CB Y j SE+ c a a -2 1 l -; a 1 . £ 3 ¢ = n &
λ 5.11.2. Sums containing Cm
1.
2.
3.
4.
5.
6.
9! ' c 0& ' % P 5 1 J
5 $ ¢ 43 ')( $ ! % 5 J
( ' (')( (*')(
(87 1 ¤ F G &N
7( ' (% V( ')(* ! 1;- L' &! 1 (9! ' & " ¢ 3h J ( PL 5n c & ' 5 6 ' _ 5 7 ! % "
1L" 1L' J
= '*>?7R@CB
( 9! ' & $ 5 9!& '! " ¢ ')( $ ! % (Q(*'65_7X! " (P5n'65_7X! " c 0& ' % ( &! ' & ¤ 4 3 $ 5 P5_7X! & L" ¢ J
43 ')( $ ! % 5 1 L " ( P5n'65n O! " c a 1 & J
£ d2 ( 5 7! ' & ¢ 3h ¤ ' % 5 1L' J
( ( 9 ! C 7 ( * ( 9 !" " ¢ ' £) £! ')( (P5_7X! " S c & $ L 4 3 3 d 3 -
J 3 £ 3 £ d2 43 3 d2 a ¢ 3h c a & = > O'I@B S - 1 Q(*'! " C7 (* Q(*'! " 43 ¢ $ L 0d 3 £) ')( $ ! % ( (* c 0& ') ( P5_7X! " &-
J 3 £ S 3 £ 0d2 4ST3Pd 3 4SEa ¢ 3h
S c a2S & = 'I@CB 1 &- S
λ 5.11.3. Sums containing Cm
1.
pk (z)
nk pk (z)
")" ' & -21 ')( $ ! ')( $ ! % C7 (& ! " 63 ¢ - c - 0& ( ('! " Q(*'! " (:7 ¤ ' J % C 1 7( L &' ! F 7(8 G F G ' (*'65 1 Q " L J
3 :;
.
; 3
7 $ ')( $ ! C7 (* &! " ')( $ ! % C 7(%9! " F 7(8 G c - & F 3 d G ( 7 ) : 1
2.
3.
4.
5.
6.
7.
8.
m !" C7 (& ! " $9% )" " C7 ( &! "
,
= '*>?7R@CB
( ' m (
7! ' 1 ! £ - ¢ 3h& - c - & m 5_ ¤ ' % &N
m 5 7 # 7( 1 -
¢ 3h& - c - & 7! % . 5 2 CP7 5_(87 ! D ¢ 3 ( '6 5_ &! '%& j + -a c -21Rl c -;&-!&4 F E(:7 GIH 1 (9! " S a2S $ 5 ! % c a 0& &"( % C7 (&! ( S 43 ¢ $ L c - S F $ G SEa J ' (9! " $J L P5 ! " £ 0 ¢ S c a - 0& 7 S 9! ( F 5_ a &-21 lC- c - S -; . F 7<5(8 G G j + c S 5 1 ( L J
('])" "'& C7 ( &! " c - 0&
" 1 (*' &BB B ! 3 ¢ ( &'! '! % £ & SEa N S ( ' 7 ( Q(*' #7 &B BB #7 1
¤
¤
¤
¤
&L ' J 1 L" C7 ( &! " c - & 3 C7J (
&! '%& c -;a &1 -21 & ¤
' & -21 ¢ ')( ')! ( C$7 ! (9 ! $ % 9. " c - 0& 3 ( &! ' 3 ¢ ' % 1L' J
" ;5 7! ' ')( $ ! % C7 ( &! " c - & C7 (&! ' 10.
3 : :
('! " J 1 (*' L" C7(%6 (*'! " -
c -;
) *
I5_7 , ¤
= '8>?7A@ B
0 ' /
11.
7 ¢ ' ' 2- 1 ')( $ ! % C 7 (&! " c - & $ $ L L 4 3 J J
12.
13.
14.
15.
16.
17.
18.
O' '% ! % N ( ' # 7 ( ' ¤ &
1
!
m !" ¢ '*( $ ! % ! " C7E( 9! " c - 0& 4 3
( ' m . ' % ( m ! ! ' N ¤ ! ' & m ( ( ' 5 7 1
1 & L' J L" ¢ C7( &! " 3h
- c - & 3 # ('J 9 ! ' & ¢ 3h
-; c -;a & ¤ 1
( L' 7 ¤ 1 ¢ J ')( $ ! % C7 ( &! " 0 3 - c - & ' % (:7 ,
1 L' ) J
* ( ' 1 L " ')( $ ! % C7 J ( &! " C7 ( 6( '! " - c - &
( ( #$ 1;- 1 ' J % 1 ( &L ! ' - N ¤ '
1 1 &
&
! 1L" ¢ ')( $ ! % J m ! " C7(&! " c - & 4 3
7 ! j + c a !- &4 lC- -; a 1 . £ 3 ¢ ¤ m ' " ( " '#" (:7 L " 1 L " ' J1 ¢ ¢ C7( & ! " ')( J $ 5_7X! 0 3 - c - & 43 '65_7 J $ L
&L ' . £ ¢ ¢ Y ] C7 J ( 9! ' & c -;a & -21 0& 43 j + c a -;&!- &4 lC-;&!- &4 3 ¤ 1 ¢ C7 ( m ( '! " ')( $ ! % C7(&! " - c - & 4 3
( ' 6( ' #$ !'
-Q1 a 1 ¤ ' % C 7 m ( 9! ' - N & m m ( 1 1 !
3 :<
.
; 3
( P 5_7 $ 1 ( L " 1 L " J ( Q(:7X!A $ (* P5 ! ')( $ ! % ' (% 5 J C7 (9! " 19. $ & J (9! ' & L " # C 7 Y ¢ 3h
- c - & 3 ' % ')( Q(:7X!A ')( P5 !A C7 (&! ' ¢ 3h
-;
( ' Q(*' #7 7(8 ¤ Y N ! & c a 1 ( ' c a (*'
20.
21.
22.
23.
24.
25.
( 9! " ¢ ' a 0& $ 4 3 L c S ( J 1 )L " -
J
(9 ! ( j + a c -21 lC-;&-21 . £ 3 S 1L( J
( 9! " ' ¢ c a 0& $
L 0 3 ( J
S -
1 L" J
( 9! ( 3 0 3 ¢ j + c a -21 l -;&-21 . £
S 1 L ( 1 J
=
O ' ' ( $ J 1 7
¢ = > 'I@CB
3 ¢ > ' !
> @CB
1L " 3 ¢ S a - 1 & L $ 5 J ! % ( 1 L)" O' ' J%
7 . ' L D 5n'! % j S + 9lC-;&-21 £ 3 ¢ <3 % j S & H = >_'I@CB J ( 9! " S a2S $ 5 ! % c a &
&-
"( S % C7 ¢ $ L ¢ 3 SEa c - S F G ¤ ( 9! ( 3
SEa 7( $ J S 5n')(* $ " ( &! " ¢ ( $ L $&% &! " 3h
SEc a a & J &-
5n 9! ' ' > @CB &! ' S c 0& c & = * m !" ¢ ' C7 ( &! " c - a S & $ 4 3 L
J 3 ¢ S ( m 5_5n7X'! ' ! % ( 9! ( N ( (*( ' m ( ( ' # 5 ¤ m &
1
! 3 : L
26.
27.
28.
29.
30.
31.
32.
33.
34.
0 ' /
¢ ' J 5 1L " C 7(%9 ! " c - a S 0& $ 4 3 L
J 3 ¢ SEa ( 9! ( j + -;&-21 l c -21 . ¢ 3 £
¤ SEa 1L( J
¢ ' ( P5n 5n'! " C7(&! " c - a S & c SEa & ¤ $ L 4 3 J
m !" ' C7% ( 9! " 0 3 ¢ - c - a S & $ L
J ( # (*' m ( 1 (Q( $ C 7 ( 5 ! ( & ! ( ' m ¤ W 5n'! % 3 ¢ S & N
m ( (*' 1 ! W
-Q1 J & L " . C7( &! " c - a & £ d2 j + &4 l c -;&-!&4 ¢ 3 £
¤
1 ('])" "'& C7 ( &! " c - a &
1 ( ' ( 1 (*' 9BB B 9! ' & £ 43 ¢ ( '6 a 5_7! % & 1 SEa 1 N S '( 6(*' #7 &B BB #7 " ¤ ! " " ' & ;5_7X! c -;a ;5 7 , ¤ ')( $ ! % C7 ( &! " c - a 0& C7 ( &! '
1 ) *
1 I 1 ( L' 7 ¤ ¢ ')( $ ! % C7 ( &! " 0 3 - c - a 0& ' J% & L ' ) (87 ,
1 J
J&L" ¢ C7 ( &! " 3h
- c - a &
1 3 ¢ £ d2; ¢ 3h O -; j + c -;&-21 l &4 . £ 3 ¢ ¤ ')( $ ! ' &-21 ¢ ')( $ ! % C7 ( &! " c - a 0& 43
1 ( '! " ( 1 (*' L " #( &! '%& a 1 3 ¢ J (* '! " - = '*>?7R@CB '6 ( ' % & L ' J
3 : P
.
35.
36.
37.
38.
; 3
7 ¢ ' ' 2- 1 ')( $ ! % C 7 (&! " c - a & $ $ L L 4 3 J J
1 £ d2 O' '% ! % N ( ' 7 ( ' P 5_7 ¤ &
& !
' &L " C7(%9! " J ' ( $ 5_7! 3 ¢ - c - a & 'g5 7 0 3 ¢ -;&-21 $ L J
1 . ' L ¢ £ ¢ Y '( J 9 ! ' & c -;a &-21 0& 3 43 j + c a -;&!- &4 lC-;&-21 3 ¤ & 1 m !" ¢ ')( $ ! % ! " C7 (9! " c - a 0& 4 3
1 # 5 7 £ d2 ' % ( m ! ! ' N ( ' ( m (* ' 5_7 ! & ¤ ' & m 6
&L " ¢ ')( $ ! % J m ! " C7(&! " c - a 0& 4 3
1 9! j + c a !- &4 lC- -; a &4 . £ 3 ¢ ¤ m '
'65 7! % $&% C7(&! " c - a & -21 D ¢ 3 C7 (&! '%& c -;a & -21 0& H ¤
1 ( 1 (*' L " J ')( $ ! % C7 ( &! " ( Q(*' ! " - c - a & 40.
1 ( #$ L' 1;- P5_7 1 (% ¤ £ d ' % ( J & 5_ 7X! ' 1A- N &
& &
! ( L" &L" ( P 5_7 $ 1 J ( Q(:7X!A $ (* P5 ! ')( $ ! % ' % 41. ( 5 &J L " ( &! "'& $ J 1 ( L '% & Y ¢ 3h - c - a & 3 ' % ')(* Q(:J 7 !A ')(* 5 !A '( 9! ' ¢ 3h -;
1 (' #7 # Q(*'65_7 7 (8 ¤ Y N ! & c a 1 (*' c a (*' 39.
3<9N
42.
43.
44.
45.
46.
47.
48.
49.
0 ' /
m !" ¢ ' C7 (&! " $ L 4 3 J £ 3 ¢ S (
c - a SEa &
1 m 5_7X! ' (9 ! ( &
(
(*' m ( P5 _ 5 7 ¤ ( ( * ' m & !
¢ ' (P5n 5n'65_7X! " c ¤ c - a E C7 (&! " $ L 4 3 & & E S a a S a
1
1 J ¢ ' J 5 & L)" C7% ( 9 ! " c - a SEa 0& $ L 4 3 J
1 £ 43 ¢ SEa ( 9! ( & j + 1 -;9l c -21 . ¢ 3 £
¤ SEa &L ( J
' n 5 ! $ % C7(%9! " ( P5n ( 'g5 7! " £ - 0 3 ¢ - c - a SEa & $ L J
1 ( Q( ! ' ¢ ( Q! % (*( 9'! ! ( ( C 7 '( 6 (* ! ' 3h
-; c S -; & ¤ 5: $ 5 7! % ' C 7 ( % 9 ! " ( P5n ( 'g5: O! " £ - 0 3 ¢ - c - a SEa & $ L J
1 5_7X! % ( 9! ( & ( 6( (Q(*'! ( ('6J (* 1 (:7 ! L ' ¢ 3h -; c -; 0& ¤ & '
SEa 1 m !" ' C7(%9! " 0 3 ¢ - c - a SEa & $ L J
1 ( ( Q( #$ (*' m ( 1 C7 ( m 5 5n! '' ! ( %9! ( & £ ;0 3 ¢ S N ¤ W &
m ( (*' & ! W
-Q1 ( &! " S a 5 ! a2S $ % c &- a 1 & "( S % C7% ¢ ¢ ¤ ( 9 ! ( 3 J $ L 3 SEa a 1 c SE- a S a 1 F 7( $ G ( 9! " ¢ ' a a 0& $ L 4 3 c S ( ( J 1L" - 1 J
£ ( 9! ( & . j S + a c -21 lM1 -; £ 3 ¢ = > 'I@CB & ( J L 5n'! %
3<
&N
.
50.
; 3
(&! " ' ¢ c a & $ L
3 ( ( 1L" J
S - a 1 J
£ (&! ( & 3 3 ¢ j + c a -21 l -; a 1 . £ 3 ¢ S & L ( 1 J
= > ' ! > @CB O' 1L " ¢ a 1 0& ' ( $ L $ 5 ! J% ( (
3 J
S - a 1 1L" 1 7 O' J ' L D ' 5n% '! % j + 9 lM1 -; . £ 3 ¢ <3 7 % j SEa & H = > 'I@CB S
1 J ')(&! " (& ! '%! & 0& ¤ 3 £) d2 m $9(% C97 ! ( " ((* ' ! c 0 & " m m ' &-
WX\ X\]W . (:7 5n ' & 5n')( $ G &c !- & 0& ] d j + l F G ¤ $9% F3 3 ¢ ')( $&% $ ! % d2 7(* c a 0& ,
)
&- (' V(')( 1 V('65 1 #$ &! ' F G N ¤ (
' 8 ( 7 # 5 1 (* "
& 1 J L'
! 43 ¢ ')( $&% $ ! % d2 7(8 c a & ,
)
&- a 1 " ' (' V(')( 1 V(')( 1 #$ 9! '%& a 1 N " ¤ & ( O')(:7# 5 1 ! (*
&L ' J
$ 5n Q(:7X!A m ! " c -Q1 " ( &! " J 5 L (*' ¢ 3h
c a 0& a ( : 5 g ' & $ %
&-
c & m L" 1L " 1 L " J J J
( ' ]'65 #$
c a 1 ( m Q (: O'7X!! % ' & N ¤ & c a 1 # 6(* m 5 1 ! 7(*
$ 5n 6(87!A m ! " c -Q1 L " ( 9! " ¢ 3h c a 0& J5 ( : 5 6 ' ( ( ' a 9 $ %
&- a 1
c & mL " 1L " 1 L " J J
J
( ' ]'65 P5_7 #$
c a 1 ( m ¤ Q (:'675_! 7'%! & % N & c a 1 Q(* m 5 1 ! 7(*
3< ,
51.
52.
53.
54.
55.
56.
57.
0 ' /
5 7 $ (:7X! c -Q1 L " (9! " a ¢ h & 3 ¢ J & c & 3
5 '65 1 L " 1 (*' L " 9$ % Pn
&J
J
(' 5 ' #$
c -Q1 # : 6 (8 '7!! % '%& N W ¤ a & c -Q1 c 1 ! &- &
58.
59.
5 7 $ (:7X! c & 3 ¢ 5n'65 1 J $9% P L " J
J
-Q1 L " (9! " ¢ 3
& c &a ( 1 (*' L " h
(' 6(87! '%&
c '65_7! % & N
c -Q1
λ
5.11.4. Sums containing Cm 1.
2.
3.
4.
nk pk (z)
a 1 & #$ -Q1 # 5:'65_7 W ¤
c a 1 &- &
!
and special functions
']" " (:7! ')(* $ ! % C7 (&! " #& - &- c - 0&
]! " ' 3 ] C7 (% 9! '#" c - a 1 & = '*>?7R@CB
& -21 " " " ' ]! ]! ')(* $ ! % '( 9! "'& &- c - a 0& 3 # '( 9! ' c - a 1 & ¤
1 m 5_7! ' m !" C7 ( &! " fI c - & ' %
7 Y D 0f$ ¢ <3 m H N &
' I 3 m 5_7X! N &
( ' m ! m 5_7 1 7( ' m 5 7 m 5_7# 5 7 m 5n m 5n & !
2 m 5_7! ' m !" C 7 (&! " If c - a 0& ' %
1 7 ( ' m P5_7 Y D 0f$ ¢ <3 m H N m 5_7 ! ` & ^ 7(*' m 5_7 m 5_7# 5n ', (P5_7! 3 m 5_7X! N & m 5n m 5n !
3<93
¤
¤
.
; 5
)" " M (87! ")" a ')( $ 5_7 ! % C7 (%9! " 0 fg3 c - & F 7(* G 1 5. '%& ( L ' & ( 1L 1 ¢ Y 0fg3 3 '65_J 7 ! % C7(* &! '%& 3 CJ7 ( 9! '%& c a -;&-21 0& 1 ( ' ] 6( ' V7#7 #$ (1 L ' ; 1 ! ' % C7 (*J 9 ! ' m ( ' (87! F 7(* G N & Q(*'65 1 m (*'
C7 ( m ! " ")" " C7 ( &! " M (87! 0fg3 c - & 6. ( C7 ( ! ' ' % CJ 7 1 ( &L ! ' (*')m (:7X! F (8 7 G ' m ( ' ] 6( ' ( ' (87 #7 Y fg3 3 ¢ fg3 3 ¢ N Q(*'65 m (*' &
1 m ! 1- (' Q(*' m (*')(:7 m (*')(:7 N & Q(*'65 1 m (*' m (*' 1- !
('! " ¢ C7 ( &! " F 3 7.
3 G j &- Q c - & ( ' (' # £ J 1' L % ' 3 ¢ N W W " & 1 (*' 1 ! ( (:7X!
('! " ¢ C7 ( &! " F 3 8.
3 G j &- Q c - a 1 & ( ' (' # 5 7 £ a 1 J 1' L % ' d2; 3 ¢ N W W " * ( '
& 1 & ! ( (:7!
_ 5 X 7 ! $ & J L (* " ' 0 3 ¢ j Q a &4 & 9. : 5 g ' 5 7 ! $ %
&-
1 J L" ( ' ]'65 & 1 'g' 5 % 7 N !A C7 (* ! & 7 #7 ! C7 ( $ 5_7X! & L " ¢ j Q 0 a &4 0& J
3 10. $ 5n'65_7! % ( 1 (*' )L "
&- a 1 J
( ' ]'65 1 '6'5 % !# N !A C7 (8 ! & 7#7 ! C7 ( 3
¤
¤
¤
¤
¤
¤
11.
12.
$ 5 7!A $ ! % ¢ 3h O j ' 5 & L " 1 (*' L "
) * 6 J J
Y a 1 & £ ¢
&-
$ 5 !A $ 5 7! % ¢ j a 1 h 3 O
* ( ' 6 ' 5 ) L " 1 L " J
J
O'Q5 ! Y a 0& O'! % % ¢ 3
&-
13.
0 ' /
$ 5 !A $ 5_7! % ¢ 3
' 5 L " ( 1 (*' L " h 6 J J Y a a &
&- 1
7 (
N & 7
*
N &
j a 1 ) * ¢ X £ X £ Y
N &
7<5
,
#$ ( ' ] '65_7 1 & 1 1 V7 ! C 7( !A C7 (8 !
( 7<5 , j a 1 * , ) #$ (' 'g5: & 7 & & C7 ( !A C7 (8 !
! 7( 7,5 , j a 1 * , ) ¢ 3 #$ (' '65 & 7 & & C7 ( !A C7 (8 !
! 7( 7<5 , a 1 ) * , j a 1 ¢ 3 R ¢ 3h
, j ) *
¤
¤
¤
& J 7 ! L % " (*' ¢ 3h O a n 5 6 ' _ 5 $
1) * 1 J L " ( 7! ' ¤ Y a &4 0& 7 * (
&-
1 J L' & J 7 ! % L ( " (*' ¢ 3h
15. n 5 6 ' _ 5 $ 1 )L " 7( J
7<5 a O'65 ! ' 7% ( Y a a
4 & 0 & , 1 ) * , &- a 1
1) * #$ ( ' ]'65 Y N ¤
1 & ! C7( !A C7 (8 !
( g ' 5 1 L)" ')J( $ ! % C7 ( &! " F E(:7G &- Q 16. ( ' ] Q(*' W W #$ ( 1 1 L ' L ' - ¤ ! Y c - 0& £ &A J ' % C7 J (* &! ' 63 ¢ -; N 6(*'65 1 1
14.
3< ;
.
17.
18.
19.
20.
21.
22.
23.
; 5
( ' ( 1L" J ')( $ ! % C7 (&! " F E(:7 G &- a 1 Q c - & (' Q(*' W W (% L ' & 1 ' L - £ &A a 1 J ' % M7J (* 9! ! Q3 ¢ -; N
Q(*'65 1 &
'
* ( ' 1 L)" ')( J $ ! % C7 ( &! " F G &- Q c - & ' ( 1;- 1;- &<- " " ' ! % N 1 7(% ! ( 1 (*' L)" J ')( $ ! % C7 ( &! " F G &- a 1 Q c - & '& ( V( 1 a 1;- 1- " " '! % N 1 7 (
! "" 7 (8 ) O')(* $ ! % C7 ( &! " &- Q c - &
C7 (8 " ! ( ' (*' (1 L ' 1 J
¢ !
6( 'g5 1 1 ' % 1 L ' 3h
-; N
""J 7(* ) O')(* $ 5_7X! % C7 ( &! " &- a 1 Q c - 0&
C7 (8 " ! (' (*' (1 L ' 1 £ J
¢
6(*'6! 5 1 & ' % 1 L ' 3h -; N
"" J 7 (8 ) O')(* $ ! % C7 ( &! " &- Q c - a 0&
1 C7 (8 " ! (' ( ')( (1 L ' 1 ¢ £ d J
! a
5 1 1 Q(*'6 ' % & L ' 1 3h
-; N
J " "
7(* ) O')(* $ 5_7X! % C7 ( &! " &- a 1 Q c - a 0&
1 C7 (8 " ! ( ' ( ' ( (1 L ' 1 d J
¢ ! a
5 1 & Q(*'6 ' % & L ' 1 3h
-; N
J
3<9:
#$
¤
¤
¤
#$
#$
¤
¤
#$
#$
¤
¤
24.
25.
26.
27.
28.
29.
30.
0 ' /
( Q(*'! " £ C 7( ! " 3 & - c Q - & & ' ( ;1 - V( c a 1- c - " " C( ' % ! N 1 7 ( ! ( Q(*'! " £ ¢ C7( ! " - 3h& - c Q - 0& & ( ' ] )(*' #$ ( P5_7! ' 1 ( ' L - ' % C7 (*J ! ' F 7 (8 G N 5_7 ( ! 'g5
1
( O! )" " C7(! " F (:7 G c a Q - & &( ' J % C 1 7 (* L' ! F (87G N ( ') (*( '6Q5 (*' )(*' ' & 1 1 ,-Q1
! ( Q(*'! " ¢ C7( ! " 3h
- &c - Q - &
( ' (*' ( 5_7X! ' 1 ( ' C7 (8 " ! #$ L 1 ¢ ! 3 -; N
P 5_7 )(*'65 1 ' % J 1 L ' J
( Q * ( ' ! " C7( ! " ¢ 3h
- &c - Q - a &
1 (' V(')( ( P5_7! ' 1 ( C7(* " ! #$ L ' 1 £ ¢ ! a 1 0 3 -; N
5_7 ( 'g5 1 ' % J& L ' J
( P5_7X! 1 ( $ a L"J1L" J
- c 1 Q c - & G
F 8 ( 7 ) ' ( 5 C 7 ( & ! "
&& L" J
( L ' & J C 1 7% ( 9! ' - c F 7(8 G ( L" &L" a $ ( P5_7X! J 1 J
c 1 G
F Q c 8 ( 7 a ) ' ( 5 C 7 ( & ! "
1 & && L" J
9 1 ( L'& J ( &! ' - c -21 F 7(8 G 3< <
¤
¤
¤
¤
¤
¤
¤
.
31.
32.
33.
34.
35.
; ;
7 C7(! " F 7(* G c a Q - &
&(1 L ' J '! % , N 7 8 ( & 1 ) 7 C7(! " F 7(* G c a Q - a 0&
1 & 1 ( L ' J '! % £ & a 1 F 7(* G N & 1 43 Q - c - Q a 0 & &-
2.
3.
( ' #$ 1 * ¤ \] W -Q1
(' V (')( (*'65 1
!
(' V(')( V(')( 1 #$ ¤ \]W )(*'65 1 -Q1
!
( #$ 1;- V ( ¤ ' ! % ' £ & N & 1 7 ( )(*' ! ( 1 W W ( ! ! ' " 1- ! (P5_7X! " c Q a 0& ' % £ & N ¤ _ 5 7 # 7 ( (
'
&-
7X! ' £) 3 3 d2 R43 3 d2 c - a ¢ c 0& 3 ( P5_ ¤ ' % & &-
λ 5.11.5. Sums containing products of Cm
1.
( P5 $ !A( 5n'! " ¢ ')( $ ! % (Q(*'65_7X! " P5n'65_7! " 4 3 9! ' & ' % ( &! '
nk pk (z)
c &
1 ' J 5 L c 0& = = @ !C@CB
1 J L'
$ 5 9 ! $&% 5n')( 1 L)" ¢ ( ' L J " 9! " P 5n'65_7X! " c 0&
43 ')( $ ! % & * J £ d, ¢ 3 £ (9! ' P 5_7X! ' c -21 0& 5 1 Q(:7X! ' L' J
$ 5n Q(:7X!A Q(:7X! " (:7 a , S c - 0& $&% 5 1 L " 5: $ ! ( )" " P5n $ ! '#)" " ) J
7 5:' 9Q! (: > ' ( &(' J ' L SEc a & = 3<9L
= = @ !C@CB
!
c&-a 0 & = @ !C@CB
0 ' /
! " ¢ C7 ( &! " h c - Q a a & 3
&- 1 ( ' ( ' ( 1 # #$ ! ' & a 1 O '6 5_7X! % & N g 5 1 1 C7 (8 " !
! ! " ¢ C7 ( &! " 3h
c - Q a & 5.
&-
#$ (' 1 ( ' O'! ! %' N " & 65 1 1 ! C7(* !
! " ¢ a C7 ( &! " 3h c - a Q 6.
1
&- & (' 1 ( ' P5_7 #$ £ d Q O'! ! %' N " & g5 1 & ! C7(8 !
! " ¢ a C7 ( &! " 3h
c - a Q 7.
1
&- a 1 & ( ' ( ' ( #P5_7 #$ ! ' & £ d Q a 1 '6 1 C7 (8 " ! 5_7X! % & N g5 1 &
! $&% ( 5 $ ! 9! " P5n'65_7X! " c a Q c &
8. & & !' '%% & N ( ' P 5 ! (
1
$ 5 &! $9% P5 1 L " ¢ c a a 1 Q c 0&
J
9. 3 9! " P5n'65_7X! " 1 ( ' L "
&-
J
#$ (' # P5n'65 1 5_7! ' 2 P O'! % & N 5 C7 ( !A C7 (8 ! 1
! $ 5 9! $&% 5 1 L " ¢ c a a 1 Q c &
J
3 10. 9 ! P n 5 6 ' _ 5 X 7 ! ( * ( ' " " 1
&- a 1 L " J #$ (' # P5n'65 & !' & & '6 5_7X! % & N 5 C7 ( !A C7 (8 ! 1
! 4.
3<9P
¤
¤
¤
¤
¤
¤
¤
.
11.
12.
13.
14.
15.
; ;
& (* L " £ 7( J * - ) 7(* , &--
c 1 ( 'L" J
(' ')(* P5 C 7 * ( 9 A ! ( 9 ! ' & Y c - 0& '! % N& 7( #7(*
!
( 9! " $ ( P5_7X! $9% C7 * 5 O! " C 7 (&! " ')(* :
( 9! " & (* L " 7 ( $ ( P5_7X! $9% C7 * £ J
C 7 (&! " ')(* 5:O ! " ( 1 (*' L " - ) 7 (8 , &J ( ' ]')(* 5 ( 9! '%& Y c - &
] O'Q5_7! % & N 7( #7( ( P5n $ !A $ ! % J P5 (P5n'65_7X! " 1 ( ' L J
a Y c a 1
&-
"
1L "
9! " ¢ 3
O
a &4 Q 1 ( #$
WW ¤ 1;1;-
c a &4
a 1 Q # 1 (W $ ¤
W ! 1;1-
7(8 7,5 Q c ) * , c ) * , ( ' c c a 1 #P5n'65 1 ]& O! '' ! & % N !A C7 (8 & # 5 1 1 ! C7 (
(P5n $ !A $ ! % P5 1 J (*' L "9! " ¢ 3 O ( P n 5 6 ' _ 5 X 7 ! ( " 1 J L " 7(8 7,5 Y c a a a 1 Q c c , ) * ,
) *
&- 1 ( ' c c a 1 # P5n'65 & !'& # & '6 5_7! % N & # 5 C7 ( !A C7 (8 1 1
! (P5n $ 5_7!A $ 5_7! % J 5 & L " ¢
3 (P5n'65n ! " 1 ( ' L " 5 7! " J
7(8 7,5 a Y c a &4 Q c a c a , !1 ) * ,
!1 ) *
&-
( ' c a 1 c 5_7 #P5n'65 & (P5_ 7X !A' ! % P5 ! ' ¢ 3h N
7 & C7 ( !A C7(*
& 5 1 P5_
! 3 L N
#$
! ¤
#$
! ¤
#$
! ¤
16.
0 ' /
( P5n $ 5_7X!A $ 5_7! % J 5 & L " ¢
3
5 '65n ! " ( 1 (*' L " P5_7! " (Pn J
7(8 Y c a a a &4 Q c a
1 ) * , c a 1 ) *
&- 1 (' c a 1 c P'65n5_ O7! ! %'%& ¢ 3h N
7 &
& P5 1 # 5_
λk+µ 5.11.6. Sums containing Cmk+n (ϕ(k, z))
1.
2.
3.
4.
5.
6.
7.
8.
!
7,5 , 5_7P5n'65 #$ ¤ C7 ( !A C7 (8 !
¢ ' ¡ = 'I@CB 4 3 J $ L S c & ¢ ' S c ¢ & $ 4 3 L J ')( ! " P5 5n'! " ' S -; ¤ % &! ( & ' , G ,G F 3 F 3 a (*'! % P5 1 L ' $ 5n'! % 5n'65 1 L " J J
' ¢ 3 J $L SEc a & S '! " £ a a 0d2 3 £ & ( $ 5n n 5 '! % & S c Q ¤ a (&! ( ¢ ' S S c F ¢ $ G 3 ¢ S £ d2 4S S 3 % £ & S = '8> @CB $ 4 3 L l J 1 ¢ ' ¢ S £ d2 S S 3 9! % ( S S c F $$ 5n * ( G $ L 4 3 h 3 & 3 S l J 1 = '*> @CB (9! ( % ( &! ( ¢ ' c S / 43 ¢ S ! % £ & 4 S S l 3 43 ¢ S % $ L 4 3
J 1 = '*> @CB % ( 9! ( & ¢ ' c SEa / 3 ¢ S 5_7! % £ & SEa 1 S l $ L
4 3 2 1
1 J #( &! ( & '8> @ B 1 3 43 ¢ S = % ( 9! ( ¢ ' S '8> @CB c S F $ G d2 S S l 3 ! % £ & S $ L 4 3 =
J 1 3 L1
.
¢ ' ES a c SEa F $ G £ 0d2 SEa $ L 4 3
1
1 J 1 3 % ¢ ' c S / ¢ ! % 0d2 S $ L 10. 4 3
J 1 ( 7! " ' c SEa / ¢ 3 ¢ S $ L 11. , 7 5 $
1 J 1 9.
; :
1 S l (&! ( & 5_7X! % £ & SEa 1 = '8> @ B &! ( S 43V& S l 3 ! % = '*> @CB ( & % 5_7X! % 0d2 SEa 1 S S l 9! ( & 3 5 7! % = '8> C@ B
¢ ' $ L S c F G 3 ( & ! ( ! % £ & S
S $ J S -; (P5 ! ( "' £ S O')(* ! " (* '! % ( P5_7! " £ & -
& - a $ 5:'! % ')(* = '*> @CB ¢ ' SEa c SEa F $ G $ 4 3 L
1
1 J ( &! ( & ' % ( &! ( & ( P5 5_7! ( "' £ S 1 (* '! % 3 5_7! % £ & SEa 1 & - a 1 S ')(* ! " Y -; a $ 5n'! % 'T(* ( &! " (* ')(* $ 5_7! £ & - = '*> @CB ( 7X! ( % £ ¢ ' S ¢ c S #F " $ G ! % d2 S S l
43 J $L ( 9! ( 1 3 ! % V& S = '8> @B ( ¢ ' $ & ¢ G 43 J $ L $ 5n ( c SEa 1 F " $ 1 ( & £ SEa S '8> @ B ( 7! 5_ 7X! % % £ d2 SEa S 3 ( &! 5_ 7! % 1 =
1 l $ ¢ ' 43 ¢ S % ! % £ d2 S S S $ L 4 3 & c S * $ 5 ,
l
) J ( 9! ( 1 % 3& S = '8> @CB 3
12. 43 ' % (&1 ! (
13.
14.
15.
16.
3 L,
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
0 ' /
$ ¢ ' & SEa 1 c 5 , $ $ L 4 3
2 1 E S a
1) * J 1 3 ¢ S 5 % 7! % £ d2 SEa S 3 # (&! ( % & 3P& S = '*> @CB
1 l ¢ ' d2 S S 3 &! ! ( % S S 5 $ $ 4 3 L & c S , l
) * J 1 = '*> @CB ¢ ' SEa 5 , $ $ L 4 3 & 1 c SEa
1 )+* J &! ( & 1 43 ¢ SEa 1 £ / 0d2 SEa 1 S l 43 ¢ S 5_7! % SEa 1 = '*> @B 7 $ 7 ¢ ' S S c * " $ , 43 ¢ S d2 SQ - S S $ L 4 3
l ) J (9! ( S 1 8 ' 3 % = > @CB &! '%& ¤ $ "" a ¢ £ ) ' : ( 7! % P5_7X! 9 $ % c 0d2 3 & &- & 3 1 m $ m 5 ! '#)" "" ')( $ ! % C7 ( &! " F O G c - ¢ & ' L ' 5 J !A C7 (&! c -; F ¢ 3 G ¤ ' m m " ) " ¢ ')( $ ! % C7 ( &! " f &- -21 c - f & ' 1 ( ' % ' 5 J 5_7X!A CL 7 ' (* &! ¤ ' m ! )" " ¢ ¢ O')( $ ! % C7 ( 9! " f &- -21 c - 4 f 4& " ' O'! % J ' 1 L ' 5_7!A C7 (%9! ¤ ' m (&! ']" $ ']" c a F $ G O')( O! % £ & &-
& $ % = '8>?7A@CB 0 2 d
&-
1 (&! ' $ ']" a '8>?7A@CB c a F $ G ')(:7X! % £ & &-21 9 $ % 2 d =
&- 1 1 3 L 3
.
27.
28.
29.
30.
7X! ¤ 1L " a ¢ 3 '6')5_ J
1 : ( 7X! % G
F
$ n 5 '! % 1 (*' L "
&-
1 J
" 1L " ¢ a J
1
F G , 7 5 $ a 5 n ' ! ( ' ( $ %
&- 1 1 L" 1 J
3 '6')5 (:7X!! % ¤ " 1L " a 1 F ¢ G
$ 5nJ ' ! % * ( '
&-
1 1 J L)" 7 O'65 ! 7! 3 '65_ D ')(:7! % ')(* O! % H ¤ " 1L " J a
$ 5n'! % ( ' ( , 7 5 $
&1 L" 1 J
3 '65 1 J ! % L " (*'
n 5 ' $ 1 1 J '65 L
6 ' _ 5 7 A ! 3 # ')(
31.
" !A O'Q5 !%
33.
34.
1
a 1F !
" 1L " a J
1
, 7 5 $ n 5 ' ! ( ( ' $ % 1 L"
&- a 1 J 5 !A O'Q5 ! # '65 3 '65 !A #'6 ') ( !% 7 3 ( 1L " $ " J
(*' a 1 F ¢ 5 $ n 5 ' ! $ % m
&-
1 L" 1 "m J
"m 3 ' % ! 3 '65_7X! % ')(:7! N &
$ 1
¢ G
7 O'65 ! D ')(:7! % ')(* O! % H ¤
a 1 F ¢ G
&-
O! # '65_7X!A O'65 ! '65_7X! ¤ 7 ')( O! % 3 3 ')(87! %
32.
; :
F1 ¢ G !A O'65 O! '65 ! ¤ ')( O! % 3 ')(87! %
G
( ' : ( 7')( m ( m m !
" $ " 1 ¢ a " L 5 m $ 5n'! J % ( * (1 ' L " 7<5 $ &- 1 a 1 F
"J
" ( ' (:7'65 3 m ' % ! 3 '65_7X! m % '65_7! N m ( m m &
! 3 L95
V7 1 3 ¢ ¤ G V7 1 3 ¢ ¤
35.
36.
37.
38.
39.
40.
41.
42.
0 ' /
$ 5_7X! "" 1 L "'& J (*' a &4 ¢ 5 n 6 ' _ 5 X 7 ! $ %
&-
1 L" J
$ 5_7X! "" & L " " J
5 $ 5_7! $ 5n'65_7X! % ( 1 (*' L)" 7,? J
Y a &4 a
&- 1 $ 5_7X! "" 1 L "'& a ¢ J
&4
$ 5n'65_7X! % 1 (*' L "
&-
J
O'5 % £) ¢ # ' % ! ¤
¢ !£ ¢ O '6' 5 % ¤
£) ¢ #'6 ' 5_% ! 7 D ¢ £
H ¤
$ 5_7X! "" & L " " 5 '65_7X! % ( J 1 (*' L " 7,5? $ 5_7! $ n J 7 ] O'65 O! Y a &4 a ¢ £! ¢ £ D ' % # ')(:7X! % H
&- 1 $ 5_7X! "" 1 L "'& a ¢ £) ¢ J
&4
$ 5n'65_7X! % 1 (*' L "
&-
O ]'6 ' 5_% ! 7 J D ¢ 7 £ 4 ¢ X £ R £ # 4 H $ 5_7X! "" & L " " J
$ 5n'65_7X! % (')( 1 )L " 7,5? $ 5_7! J Y a &4 a ¢ £) ¢ 'Q5 O! O'65 O!A '65 ! £ &- 1 ' 7 % #7 O ') (:7X! % ] ')(* ! % & L " $ 5_7X! "'& $ 5n'65_J 7X ! % C7 (*'! " 5 7 ! 5 $ m #$ (' #7 '65 O 6 ' _ 5 7 m & ! Y a &4 ¢ £! ¢ ' % C7<5 ! N m &
&-
&- & a
" & L " $ 5_7X! "'& J
$ 5 7! 5 m $ 5n'65_7X! % ( 1 (*' )L " 7,5? $ 5_7! J
(' 'g5 V7 #$ 'g5 a ¢ ! £ ¢ Y &4 a ' % C7<5 m ! & N
&- 1 &- & a ! m
3 L;
¤
¤
¤
¤
¤
.
43.
44.
45.
46.
47.
48.
49.
; :
"" " $ 5 9 ! (9! " J P5 1 L " ' 5_7X! " ( 1 (*' L " 7,5? $ 5 9! &$ % ( 5n6 J " ' P5_7X! ' & ¤ Y c a a a 1 ¢ £) d2 ' % & L '
&- 1 J
"'& $ 1L " a ¢ J
* ( ' n 5 ' ! % 1 L)" &- 1 F G $ 1 J O'65_7X!V * ( ' ] 6 ' 5 ')(* ! % £ & N ` & ^ 7 (*' 'g5 & ` 3 ')(:7 & N ^ = '*> @CB 1L " $ "'& $ 5n'! J % ( (*' a 1 a F ¢ G , 7 5 $
&- 1 1 L" 1 J
* ( ' ] 6 ' 5 O 6 ' 5 V ! ')(* ! % £ # & N ` & ^ 7 (*' 'g5 ` 3 ')(:7 & N ^ = '*> @CB ( &! " 5 1 " L " £) a a ¢ £! d2 d2 -21 $9% ( P5n'6J 5_7X! " (*' c 1
1 L " &-
" ' J
P5_7X! ' ¤ ' % 1 L ' J
" # ' " ) " " ( ! ( ! $ $ m m ')( $ ! % C7 ( &! " c - F $ G
' ']" 1 ( ! ' m ¤ ( 7! ' % ' m ( !A C7 (9! ' c -; F " G
"$ '& ( m ! " $ m ( ! ']")"" c - a F $ G ')( $ ! % C7 ( &! " ' ']" 9 1 1 # ( 7 ! ( 7! ' m " ' & m ¤ ' % ' m ( !A C7 (9! ' c -;a FO " G
1 " $ 5 ! '#)" "" ( ! m m 'Q( $ ! % C7( 9! " _& c - ) $ 5* , *
' m ' 5 ( !Am C7 ]!( 9! c -; ¤ ' )+* m ( , m 3 L :
50.
51.
52.
53.
54.
55.
56.
0 ' /
( m ! " $ m 5 ! ]' ")"" ) ' ( $ ! % C7 (& ! " & a 1 c - a 5 , $
1 )+* ' & m " m ( ! ' & ( 7 ! m ¤ c -;a ' m 5 !A C7 (9! ' ( , m
1 ) * ( 7X! " (9! ' O ]! ' ¤ ¢ ')( $ ! % C7 ( &! " f &-21 c - F $ m 5_7 G '! % ' m 5_7X! ( 7X! " ¢ ')( $ ! % C7 (&! " f &-21 c - a F $ m 5_7 G
1
9! ' & O ]! '%& ¤ ( '6 5_7! % ' m 5_7X!
$ m 5:7! '])" "" 'g( $ ! % C7,( &! " 43& - ¢ f ¢ 4& c
)
7 75 $ m 5:7! , ( L' 1 O'J ! % ' 5n7X! F3 G ¤ m
$ m 5 7 ! ']")"" ')( $ ! % C7 (&! " 3& - ¢ f- 7 Y c - a
1 ) 7<5h $ m 5 $ 5 ¢ ' c F $ G d2 $ L
4 3 J 1 ¢ $ m 5_7X! '])" "" ')( $ ! % 9! " f ¢ 4 43
57.
¢ 43
58.
¢ 43
¢ & a 1
1 ( ' L '65_J 7X ! % ' 5_7X! ]F 3 G ¤ 7! , m ' ¤ ¢ 3 -; 3 ' %
5 7 $m _ $ m 5_7X! n 5 , ' 1 L' ( ]! ¤ O'! % J ' m 5_7! P5 1 L ' J
] ' " ' $ m 5_7! ( & ! ' ¤ ')( $ ! % C7 (9! " c - F $ m 5_7 G ' % ' m 5_7X!
1 L ' J
$ m 5 7! ']" ] (9! ' & ' & ¤ ')( $ ! % C7 (9! " c - a F $ m 5_7 G ' % ' m 5_7!
1 & J L' 3 L<
# c )
.
; <
λk+µ 5.11.7. Sums containing Cmk+n (ϕ(k, z)) and special functions
1.
2.
3.
4.
5.
('<( 1 L)" ¢ J ')( $ ! % C7 (&! " &- a 1 F ')( $ 5_7 G 3 &- -21
1 '65_7X! '#" 7 Y c - F $ G 3 ' % 6 ' _ 5 7 , a
1 ) *
( 1 (*' L" 'g5 7 £ a G ')( J $ 5 7! % C7 (&! " c - F '65_7 Q 4 1 F
( '<( 1 L" £ a ¢ 1 3 &- -21 'g( J $ ! % C7, ( &! " &- a 1 F 'g( $ 5n7
1 I 7 Y c - a F $ G 3 ' % ¢ &-21 a ' n 5 7 ,
1 ) *
1 ( 1 (*' L " '65_ 7 ¢ £ a J ( &! " c - a F 'g5 7 G ')( $ 5_ 7X! % C7 / Q 4 1 F
1 1 (*' L " £ a ¢ J ( &! " &- F ')( $ 5_7 G 1 3 &- -21 ')( $ ! % C7
9 1 Y c - a F $ G 3 ' % ¢ &-21
F 'g5 7 G
1 1 ( 'L" 'g5 7 ¢ ( &! " c - a F '65_7 G ')( $ J 5 7! % C7 / O F
1 1 (*' L " ¢ ' ( J $ ! % C7 ( &! " &- F ')( $ 5_7 G 3 &- -21 1 '65_7X! '#" 7 Y c - F $ G 3 ' % g ' 5 7 ,
) *
1 (*' L " 'g5 7 £ J G ')( $ 5 7! % C7 ( &! " c - F '65_7 3 Q 4 F
1 ( ' L" ¢ J ( &! " &- F $ 5 7 G c - a F $ 5_7 4 3 V &-21 ')( $ ! % C7
1 ( ' (87 V( ' ( # #$ '%& '%& " '65_7X! % O'Q5_7! ¢ 3 N " 1 ( & 1 1
! 3 L L
G ¤ G
G ¤
G ¤
G ¤ G ¤
0 ' /
1 ( 'L" ¢ J 4 3 V &-21 ')( $ ! % C7 (&! " &- F $ 5_7 G c - F $ 5 7 G 6.
( ' (:7V(')( # #$ ' '%& " 1 ¢ ¤ '65_7X! % '65_7!A (Q(87! N " 3
( 1 ( & 1 !
( ( ' 1 L" ¢ ( &! " 7. 4 3 V &-21 ')J( $ ! % C7 Y a F $ 5_7 G c - F $ 5_7 G
&- 1 ( ' (87 V(')( # 6(87 #$ '%& '%& " & '65_7X! % '65 !A (6(87! N ¢ ¤ " 3
( 1 ( & 1 !
( < ' ( 1 L" ¢ J ) ' ( ! C 7 ( & ! G G
$ % 8. " &- a 1 F $ 5 7 43V
c - a 1 F $ 5_7 ( ' (87 V( ' ( # #$ '%& '%& " ¤ '65_7X! % O'Q5 ! ¢ 3 N " &
( & 1 1
! 1 ( L ' ¢ ¢ $ 5 7! " a J C7(! " F O G c 4 3 Q - & C7 ( ! ' 9. & )(*'! " a ¢ ¤ Y F3 G F]3 G Q c ) ' ( 5 7 ! * ( 6 ' 5 $ % 1L" J
( ) ' ( & ! " ¢ ¢ C7 (! " &c - F ' ( $ 5_7G 10. 43 3 &- -21 1 Y - F $ G 3 ¢ &-21 c F '65_7G
(') ( &! " '65_7 G C7( ! " ')( $ 5_7X! % - F '65_7 G ¤ 3 Q F
( 9! " ¢ a ¢ (')
11. 4 3 3 &- -21 C7( ! " c F ' ( $ 5_7 G 1 & 1 Y - a F $ G 3 £ ; ¢ &-21 c F '65_7 G
1 ' 5 7 ( 8 ' ( & ! " ¢ G C7E( ! " '*( $ 5 7X! % - a F ' 5 7 G ¤ / F
1 3 L P
.
12.
13.
14.
15.
16.
17.
$ 5_7X! "" a C 7(! " c 4 & Y - F $ 5_7 G
$ 5_7X! "" a C7 (! " c 4 &
¢ Q (')(:7 )(:7 ( 9! ' & ! " ] '65_7X! % )(:7X! N V( 1
! ¢ Q - a F $ 5_7 G
1 (')(:7 ( & ! '%& ! " ¢ '65_7! % 3 N 1
! " " $ 5_7X! C7(%9! " 43 Q c a 4 3 Q c - F $ 5_7 G
& Q $ ! ( % & $ ! " 5_7X! c a & $ 5_7X! "" C7 (9! " 3 Q c a 3 Q c - a F $ 5_7 G
1 & d2 Q
( P$ 5_5n7X ! ! " % c a & $ 5_7X! "" a ¢ 4 Q c - F $ 5_7 G C7(%9! "
& (')(:7 # Q(:7 ! '& V( 1 ! ] '65_7X! % ( Q(:7! N
$ 5_7X! "" a ¢ C7 (9! " Q c - a F $ 5_7 G c 4
1 &7 ( & ! ( 7X! ' % ' & D '65_7X! % '65n O! % a /
λk+µ 5.11.8. Sums containing products of Cmk+n (ϕ(k, z))
1.
2.
; L
3 ¢ ¤
¤
¢ Q ¤
¢ Q ¤
3 ¢ ¤
:& H ¤
¢ ' ¡ = 'I@CB 43 J $ L S c & ¢ 9! ' (9! ( ¢ ' S ¢ 9G H 43 ' % S l 3 % ! #& S 43 J $L D S c F" $ 1 = I' @ B 3 P N
3.
4.
5.
6.
&' 3 465 78
( 9! " ¢ £ -21 C7( ! " 3 Q (:7! ' ! " Q ' 5_7X! % )(:7! 6
c&-a 4 ¢ ¢ - F $ 5_7 G ( ' 8 ( 7 P5n')(:7 (87 ¢ 3 N W ( 1 V( 1 ( 6 &
!
(9! " ¢ ¢ £ ¢ -21 C7( ! " 3 Q c a 4 - a F $ 5_7 G
1 & ( ' : ( 7 P 5n')(:7 (:7X! ' '6Q5_ W 3 ¢ 7X! % Q& -21 & N
6( 1 1 (
!
(&! " ¢ &-21 C7 (! " c a F $ 5_7 G - a F $ 5_7 G
1
&-
(')(:7V(')( #$ 1W ( O&'6! 5:' & O! % £ Q a
-21 ¢ 3 N
( 6 ( O ' &
1 W
! (&! " ¢ C7 (! " c a a F $ 5_7 G - a F $ 5_7 G
&- 1
1 (')(:7 V( ' ( #$ ( & ! & W '6 5 ' & ! % £ Q a & -21 ¢ 3 N
( 6 ( O ) ' : ( 7 &
1 ! W
¤
¤
¤
¤
5.12. The Jacobi Polynomials Pn(ρ, σ) (z) (ρ 5.12.1. Sums containing Pm
1.
2.
3.
pk, σ qk)
(z)
. . . j S + a l 0 & 5_7 D j SE+ a a 9lV-21 & <3 j SE+ a -21RlV-21 0& H ¤ 1 1 7<5 . ¢ ' . a G j S + -;a 9l a & j $ L + 4 3 ] F 3 = > 'I@CB & l S J 5 65 5_7X! " (:7 a . '
5 5 7! " $ 5_7! F G j S + l & J $ L '65_7X!A 5 g 5 !A C7(* ]! 5n'65_7X! . Y D - j S + -21Rl & <3 % 5 5_7% ! ' j SE+ a -21Ra lV-;&-21 . 0& H ¤ 1 3 P1
.
4.
5.
6.
7.
8.
9.
¢ ' 5 g5 5 7! " a . 5 _ 5 7! " j S + l 0& 43 J $ L ( !' 5 5_77! ! ( ( n 5 ' '65 7! ( j ¢ ' $ 5:' (87! % . 5 ! j 43 J $ L $ % S + l V- & 7 ')(:7X! % 65:5 ' ! % 5_7X! ' F (8 G j S + 9l -; ¢ ' 43 J $ L ¢ ' 43 J $ L ¢ ' 43 J $L
( ( ! " . C7 ( (*'! " j S + a l V- & 5_7 5 g5 5_7 ! ' . 2! ' F]3 G j S + ;- a 9l a &
0& = > 'I@ B
= > 'I@ B
' ( ( ! " 7 (8 a . (*6 ' 5_7! " F 7<5 G j S + l V- & J $ L '! % % 5n S + a -;9l -; . & ¤ ( ! ' ]F 3 7,5n G j E
(ρ pk, σ qk) (z) nk
2.
.
. ¤ S + a 9lV- &
. . j S + - l a 0 & j S + l a 0& = > 'I@CB ;- ( T( ! " 5 Q5 5:7! ' . . 5 5:7! " j S + a l - & 5 : 5 7! ' j S + a 9l & ¤
5.12.2. Sums containing Pm 1.
; !, ,
5 65_7! " . ¢ !£ ¢ ) ' ( ! 5 _7 ! " 5 65n'65n O! " j + l & $ % 4 3 - 5 g5 7! ' & 7 (8 ¤ '
% 5_7X! ' F G 5 g5_7! " . £) ¢ - $&% 5_7X! " j + l & 5 g5 7! ' & 5 g5_7X!A
5 g5: O! . ' % 'g 5 5_7!&7X!
+ a l 0& 5
g 5 7 I ! j # _ 5 7 ! % ' & 1 ( ' 5 g5n 5 g5n'65
5 g5 ! Y D ' % ' H ¢ 3h& N 5 g5 1;- ¤ &
5:
!
3 P,
3.
4.
&' 3 465 78
')(* $ ! % ( ')( 1 L " ( (*'! " . J $&%
j + lC- & &-
£ F G j + lC- -; . £ Q 3 ')( $ ! % . 5_7X! '#" " j + lM1A - 0& &-
' % . £ £ 5_ 7 ! j +
Q3 M l 1 ; '
43 ¢ £ 3 ¢ ')(* $ 5_7X! ' $ L ) ' ( $ 5_7!A J
¢ ¤
¢ ¤
. '65_7 $ 5 7! % ' ( $ L 5_7X! "'& j + a lC- -21 & ) J
1 ' O'65 7 ! % . £ 5_ ¢ ¤ X7 ! ' & j + alC- 1 - &-!&4 Q3 . S ( (*'! " $ 5 ! % j + l a & 6. a2S &( &(' " ( S ¤ ( 7 ! % 5n'65_7! ( 3 ¢ $ L 3 ¢ SEa j + lV- S . F $$ ((* : 7 G SEa J . . (:7 G j + - l & F E(:7 G j + -;9l a 1 0& ¤ 7. F (' m
. m 5_7X! ' ¢ m !" 5 g5_7 ¤ + l g5_7X! " j ' % & N m 5_7 g5_7 1 a & 8. 43 !
7 . 7 . ')( $ ! % m ! " j + - l 0& m ! ' j + a -21Rl - -; a 1 0& ¤ 9. 7 5 g5_7! ' 5_7 . ')( $ ! % 65_7! " j + - l 0& ' % g5 7! ' F G ¤ 10. 7 ( ! ' 5_7 . ')( $ ! % 65_7! " F 7 (8 G j + - l & ' % 6 5_7X! ' F (:7G ¤ 11. . ( 7X! " $9% g5_7X! " j + - l & 12. '65_7! % .
5 ¢ & -21 D ¢ 43 ¢ ! '%& j + a -;&-21Rl -21 0& H ¤ 1 5.
3 P 3
.
13.
14.
15.
16.
; !, ,
m !" . ')( $ ! % ! " 65_7! " j + - l & 5_7 1 a ¤ ' % ( m ! ! ' N ( ' ( m (*5 '6 g 5_7 6! 5_ 7 ' & m 7 . ' O' -21 ) ' ( ! g5_7X! " j + - l & $ L * $ L $ % J J ( ' 5 g5_7 O' '% ! % N ( ' g ¤ _ 5 7 V 7 a
& ! 1
( * ( ' ! " . ')( $ ! % 65_7! " ( ( )(*'! " F3 7,5 G j + - l & ( 5 g5_7 #$ !' 1;- V(
' % g5 5 g75_ 7! ' F 7,5 G N & a 5_7 g5_7 a . W ¤ 1 ! +1
( ! . £) " 3 I 65_7! " F 7 (8 G j + - l & ( ! '%& . 6 5_7! ' F 7 (8 G j + - &-21Rl & ¤
. 7(8 G j + - l 0& (' V(')( m 5 ( '65_7 G & N 7( ' ( m (* ' 1;- ¤ !
(')(87! " ( ! " . £) ')( 5n O! " ( ( 2! " F 7 (8 G j + - l & 18. 3 I ( ! ( ( ( ! 2' &! F 7(* a 1 D ¢ 3 ¢ ¢ ( ! ' ' & & j + a - &- l . & H ¤ G ' & 1 ( ! " . ')( $ ! % m ! " ( ( 2! " F E(:7G j + - l 0& 19. ' % ! ( ( ! ' ( ! F 7(8 G ' m ' (' a a - a 1
5 ( 'g5 7 #$ ¤ Y N a 1 (* ' (*'65_ 7 m 5 (*' 1- & !
17.
)£ C7 ( 3 I ' % C7 ( m ! ' ( ( m
m ! " ( ! " m ( ! " ( ( 2 ! " F ! '%& ! ' ( ( 2! ' F (:7
3 P95
20.
21.
22.
23.
24.
25.
&' 3 465 78
( ! " . ( ( ! " F 7 (8 G j + - l & ( ' (*' 5 (*'65_7 #$ ' % ( ( !( ' 2! F (: ¤ 7 G &N
' ( '65_7 (* O' 1;- * !
5
g n 5 6 ' _ 5 X 7 ! " . S $ 5 ! % j + a l a 0& a2S &"( S % ( ( )(*'! ( 3 ¢ $ L j + - S l - S . £) & ¤ SEa J 5 g5 7! " £ - ¢ & - j + a lV- . & 4 3 ) ' ( ! C 7 ( ( * ' ! 5 5 g 5 $ % m m " &L " C ( ! J ' 5 g5 7! '
£ ¢ ' % m ! ' m 5 5 g5 & L ' -; & -; J (' m 5 5 g5:'65_7 V( m ( ( (*' Y N ¤ (*'65_7V( ( )(*' 1 a &
!
5
g _ 5 X 7 ! " £ - ¢ & - j + a lV- . & 4 3 ) ' ( ! ! 5
( * ( 6 ' 5 $ % m " m L" J ( 2! '
7! ' 5 g 5 £ ¢ ' % m ! ' m ( ( ( & L ' -; & -; J ( ' ] m ( ( g5n')(*
5 (* m (*'65 ¤ Y N ( g ' 5 7 V ( ( * ( ' 1a &
!
5 65n'65 1 L " . J
')( $ ! % 5_7X ! " £ 0Q3 ¢ j + a l - & '! % . ' % O5_ + l a 0& ¤ X 7 ! j '
J m 5n')( 1 L " 5 g5_7X! " £ ¢ j + a l - . & ')( $ ! % m ! " 5_7X! " 6 3 5 65_7 ¤ ' 7 % N ( ' ] 'g 5: 5_m 7 (87
m ! 1;- &
3 P;
.
; !, ,
m ! " 5 g5_7X! " £ Q3 ¢ j + a l - . 0& ! 5 7 ! ) ' ( ! * ( 6 ' 5 $ % m " 1 "
( ' 1 ( m (*' V( (* ' #$ 43 ¢ m ! ' 5 g5_7X! ' Q3 ¢ N ¤ (1 m L ' 5_7! ' & 7 (* m (* ' ( ( (* ' !
J
1;- C 7 ( * ( ' ! " . m ( ( 2! " F 7 (8 G j + - lV- & 27. (' m (:7 5 (*'65_7 ' % m (! ' ( ( 2! ' ! F 7(* ¤ G &N
' m (*'65_7 ! 1;-
" ' . . ')( $ ! % ( ( ! " j + - l V- & ( ( ! ' j + -;9l -; F Q3 G ¤ 28. 7 . ')( $ ! % ( ( ! " ' ( $ 5 m ! F (:7 G j + - l - 0& 29. ( ' V7
5 (*'65_7 ¤ ' % ( ( ! ( ' 2! F (: 7 G N _ 5 7 ' m & m (*'65_7 ! 1;-
5
) * ( 6 ' _ 5 7 ! " . ')( $ ! % (*'65_7! " ( ( ! " j + - l - 0& 30. ( #$ 1;- V( ( ! . W ¤ (' % ( ( ! ! ' F 7(8 G N & ( a + 1;- ' 1;- - ( (
( * ( ' 1 m . L" J : ( 7 G j + - l - 0&
31. F ) ' ( ! * ( ' ( ( 2 ! $ % " & J L" m 5 1 L ' ( ! ' J 7 8 ( G
F m ' % ( 1 L ' ( ( 2! ' ( ' m 5 J (*'65_7 (' m
5 (*'65_7 ¤ Y £ fT3 ¢ N & m ( 1 (*'65_7 ! 1;- & N m 5 1 ( 'g5 7 ! 1-
( ! " . 'g( $
3 P :
&' 3 465 78
('#)" "'& ( ( ! " F 7 (8 G 33. ' % ( ( ( ! ' ! F (: 7 G ' 34.
35.
36.
37.
38.
39.
40.
. j + - l V- & (' 5 ( 'g5 7 9BB B SEa N S = >?7R@CB 1 (*'65_7#7 &B BB #7 ! 1;-
m !" . ')( $ ! % ( ( ! " F (:7 G j + - l - & ('
m ! ' ( ! ' 5 (* '65_7 5 g5 m (*'65_7 ¤ ' % ( ( ! ' F 7(* G & N
(*'65_7 #7 ( m (*' ! ,-Q1
( * ( ' ( ! " 1 . L" ')( $ ! J % C7 ( ( O'! " ( ( ! " F 7<5 G j + - l V- & 5 )(* O'65_7 1 a ¤ ' % ! ! ' ( ( 2( ! 2' ! F 7,5 G N ( O'
! '
' &
)(* O'65_7
1 ( (*' L " ( ! " . !$ J % C7 ( ( O'! " ( ( ! " F 7<5 G j + - lV- & ( O' 5 )(* O'65_7 ! ' ( 2! '
% ! ' ( ( 2! ' F 7,5 G & N
)(* O'65_7 ! . . m 5 $ 2! ']" 43 ¢ ')( $ ! % m 5 $ 2! " j + a ]l - + a 1 & ( 7X! ' 5n' 6 5n')(:7j + - -; a 1Rl a m m 5 $ ! ']" . . ')( $ ! % m 5 $ ! " F (:7 G j + - l - + a 1 & 5n' g7 5n')(:7 F (: 7 G j + -;9l a m $ ! % . ' 43 ¢ J $ L 5_7! " j + l - & 5 g5_7X! ' ¢ . ' %
3h& j + a 9lV-; F 5_7X! '
')( '
. . ' j + 2a - S l 0& j ES + a l -; & ¤ $ L J 3 P<
1a
¤
. -21 & ¤ . -21 & ¤ 7,5 ¤ G
.
; !, ,
m !" . ' g5 _ 5 7! " j + a2- S l 0& $ L J 43 ¢ S ( m 5_75:! ' ' ! g% 5 7! ( N ( (*( ' m (*( ' 6 5_5 7 g5 a 5_7 ¤ m &
! 1
_ 5 X 7 ! ' . "
5 g5: (*'65n O! " F 7 (8 G j + a2- S l & $ L 42. J ( !' ( ( ( (* j S + -;9 l . & ¤ : ( X 7 ! 7 * ( G F ' . . ' 5 7 G j + a2- S l V- & F 5_7 G j SE+ a l -; 0& ¤ $ L 43. F J ' ( ( ! " 7 (8 a . (*'65_7X! " F G j S + l 0& 44. J $ L ( 5 'g5 7! ( 7! ( j + -;9l . & = > 'I@ B S ' m ( ( ( 5n')(:7! " 7 (8 a . m !" 45. F G j S + - l & J $L 5 g5 (*'65_7 m 5n' 5 7% ! ( N (
= > 'I@ B &
5_7 m ! 1;-
a . ' . j + l 0& j S + a 9l V-; 0& 46. = > 'I@CB J $L S . ' j S + a l 0& $ L 47. J S F 7 (8 S a . 5 G 43 F (:7G j S + - - - -21Rl -; F (:7;G 1 = > ' ! > @CB 5_7 ¢ ' ¢ a a . 0& $ L 4 3 3 F G j S + l 48. J . 3 3 83 j S + -;a 9l & = > 'I@CB ' 5 7 a a . . G j S + l 0& j S + a 9l 0& $ L = > 'I@CB 49. F J 41.
3 P L
&' 3 465 78
' 5 g5 5_7X! " 5_7 a a . )(*'65_7! " F G j S + l 50. & J $ L ( ( ( ! ! ' j + a 9lV- . & = >_'I@CB ' S 5 65_7! " 1 ( m L "
. ')( J $ ! % C7 ( m ( '! " 5_7X! " £ 63 ¢ j + a l V- & 51. (
5 g5_7X! ' ( O' m (*' V( (* ' 43 ¢ J 1 ' % m L'! 5_7X! ¢ Q3 & N m (* O' V( ( (* ' ¤ ' m ' ! 1;-
5 g5_7X! "
£ - ¢ & - j + a l V- . & 4 3 52. ) ' ( ! ! 5
( * ( ) ' ( $ % m " m L " J ( ! '
5 g _ 5
7X! ' £ ¢ ' % m ! ' m ( ( ( & L ' -; & -; J ( ' ] m ( ( g5n' 5 (* m ( 'g5 ¤ Y N * ( ) ' : ( 7 ( ( ( ' a &
! 1
C( ( ! "
. ' £ ¢ - j + a l V- & 53. S J $L 1 ( (*' L " C( ( ( ! " - $ ( V( (*' ( ( 5 g5J 5_7X! ( 5_7 S
G % F N
( (* O' V( ( (*! 1 a = > '*>?7R@CB 5 g5 5_7X! " ' J 5n'65 1 L "
£ 63 ¢ j + a l - . 0& 5 5_7X! " $ L 54. a2S J 5 g5 5_7 ¤ 5_7X% ! ( N ( (* O' 5_5n7 O'65 5_7 7
&
! 1;-
C( ! " . a j + l - 0& 55. ')( $ ! % C7 ( m (*'! " m ( g5 1 L " 3# ( ' ] m ( g5:' ( m ( 'g5 7 5 g5_7X! ' J 5_7
( 2! '
¤ G ' % m ! ' m ( g5 1 L ' F & N (*'65_7 ( ( ( ' ! a J 1
( ( ! " m 1 L" . ')( $ ! % J C7 ( m ( '! " ( ( ! " F 5_7G j + - l - 0& 56. ( m L ' ( ! ' 1 43 ¢ ' J % ! ( ( ! F 5_7 G N ( ' (* Om ' ( ' ( O5 '6 5_(*7 'ga 5 7 ¤ ' m ' & m ! 1
3 P P
.
5 g5 5 7! " 5_ 7 ' a l a . & + G j $ L 57. F S ( ' g5 5 7! " J 1 ( * J g5_7! L " ( V( ( ' 5 g5 5 7 ( 43 ¢ S % N ( (* O' g5_7 a 1 = &
!
( 2! " a . +
l j 3 # & 58. ) ' ( ! ! ( ( * ( ' $ % m " & m )" L J
5 65n')(:7 (* m ( ! ' 5 g5_7X! ' F 7<5 G N ( ' ] (*m '6 5_7 ( ( ( ' ' % m !' m
5 ( 1L' &
! J
( (* ! " £ a a . ' $ L 59. - j + S - l & J 1 ( ( (*' L " J 5 7! ( V( ( (*' V( 5
g n 5 ( 7(8 S ! % F G & N ( ( (* ' ( ( (
60.
61.
62.
; !, ,
> '*>?7R@CB
( (*' ¤ a1
( !
= 1;- > 'I@ B
C7(* m (*'! " 1 L " J 7X! " ( ( 2! " F]3 7,5n G j + - l - . 0& ')( $ ! % C7 ( m (*'! " g5_
, 7 5 ! 5 g _ 5 7 ! ' ' m ' % ! g5_7X! F G ' m ' ( ' 1 ( m (*' ( a 1- - Y N & m 5 1 V( a a 1;- - K- ! . W
+1 a m ! " 1 L " . J +
l - 0& , 7 n 5 G j
] F 3 ')( $ ! % - a 1 L " g5_7X! " ( ( 2! "
J m ! ' ( 2! ' £ ¢ ' % 1;- - L ' ( ( 2! ' 43 -; & -; J
(' a - a 1 a K- 5_7 - Y N W & 7 ( a K- a 1 K- 5 7 ! + 1 a .
C7( m (*'! " . ')( $ ! % 65_7! " ( ( ! " F3 7,5 G j + - l - 0&
( O ' #7( m (* O'
5 )(*'65_7 ! ( ! O'm ! % ' ( ( ' ! F]3 7<5 G N ' m )(*'65_7 ! ( 1 a &
5N N
#$
#$
¤
¤
¤
63.
' J $ L J 7! ( 6 5_ !%
64.
43 ¢
65.
43 ¢
66.
&' 3 465 78
5 g5: 5_7! " 7<5
j + S a l a . 0& G F 5 65n (*'65 & L "
-
( 5
65n (*'6 5_7] 5n g5 5: N&
= > 'I@CB g5 7 5n 65 (* '65n 1 a !
')( $ ! % 7<5 . G j + l a & $&% 43 T3 F &-
(' V(')( V(')( #$ 7 ¤ 1 ¢ F (8
7 a
G & N ( ' (87 g5_ ! ,-Q1 ')( $ ! % 7<5 . G j + a l a 0& $&% - ¢ F &-
(' V(')(
1 5 g5n'65_7 ¤ 43 ¢ ¢ N (
) ' : ( 7
g5 7 ! 5_7 &
5 g5n '65_7 & - £ 3 ) ¢ 9$ % ( ( ( (* '(*! '" ! " F 3
G ( ( ( O')(:7X! '%& 7(* Y j + l . 0 & 3 ' % F G &!- & (' V( ( ' ( a a a 1 #$ ¤ Y N ( a a a V( a a & &
! & 1
- 5
6 n 5 6 ' _ 5 7 ! " 7 (8 & - ) J & )L $&" % 5n'65_7X! " & j + a l . & G F &-!& ( ' 5 g5n'65_7 #$ !' '%% & N ¤ & a &
1 ! &- &
C 7 ( ( ' ! : 5 O ) ' ( " m m 1L" . ')( $ ! % 65_J 7! " ( ( 2! " F 7,5 G j + !- & l - 0& & ! ' 5n O')( 1 L '
5 g5_7X! ' m J m '! % 6
5_7! ' ¢ & ( ' m ( #7 ( m ( ' V( (* ' #$ ¤ Y N (:7] (* 1 ( O' ( ( (* ' m & m ! a 1
67.
68.
5N1
.
5: O')(
69.
1L" . C 7 ( m ( '! " J m
) ' ( $ ! % 65_7! " ( ( ! " F 7,5 G j + -!& l - & & m !' m : 5 O')( 1 L' ( 2! ' J '! % ( ( ! ' F 7<5 G ( ' ( #7( ( ' 5 (* O'65 7 Y N (:7m ( 1 ( m ' (* '65_7 O 5n ¤
m & m !
5.12.3. Sums containing Pn
(ρ pk, σ qk) (z) mk
7 ) ' ( _ 5 7 ! $ % ( 1. ( ( 7 ! F 7 (8 '%& ( ! ' ' % ( ( ! ' m (
2.
3.
4.
; !, 3
( 2 ! " F E(:7 G a G 1 fg3 ')(:7! F 7(*
C 7( m ! " ( ( ! " F 7 (8 G ' % ( Y 0fg3 3 ¢ fg3
F
¢
3 3
0fg3 j ( ! ' C7( ( 2! ' m
and special functions
. 0f63 j + - lV- & ( ! ' & . 3 ¢ D '6 5_7X! % 3 j + a -;1 &-21Rl -;&-21 & H (' 5 (*'65_7#7 #7 G N ! 1;- ¤ ( 6 ' _ 5 7 ( ' ]
m &
+ - l m !' (*')(:7X! ('
.
0& F (87G 5 ( 'g5 7 m (*')(:7 3 ¢ N * ( &
'65_7 m (*' ! 1;- (' 5 (*'65_7 m (*')(:7 m (*')(:7 ¤ N & ( g ' 5 7 * ( ' ( ' m m ! 1;-
( '! " . G g5_7X! " j &- Q j + - l &
£ J 1' L % ' 3 ¢ N 65_7 (*' (' Va(' ( 5 a g5_ 7 (:7! " ¤ &
1 1 ! 1
. ( ! " " Ggj + - l V- 0& j F 7 8 ( C 7 8 ( ! & ' ( ! ' j + -;&-21 lV-;&-21 . & ¤ C7 (8 ! 5N,
5.
&' 3 465 78
7 ')(* $ ! % g5 7! " F (:7 G ( !' ' % 6 5_7! '
6.
7.
8.
Q j + - l . &
& -
. #$ ( ' (')( W 7<5 + ,a -Q1 ¤ ! F 7 (8 G N
1 1 (*'65_7
7 . ')(* $ 5_7X! % g5_7X! " F (:7 G &- a 1 Q j + - l 0& . #$ (' V(')( W ] ( ! < 7 5 ' + ,a -Q1 ¤ ' % 6 5_7! F 7 (8 G N ! ' 1
& (*'65_7
( ! " a . 65_7! " c Q j + - l & & (' V( 6( ' ( (*' #$ 7 ' % 5 6 g5_5_77X! ! ' 43 Q F 5_ ¤ G &N 1 ( ( ( ' (
' ! a . + 1 ('6(*'! " . ( ( ! " F 7 (8 G c Q j + - l - & & ('
( P _ 5 7 ! ( 5 (*'65_7 ! - ¤ ' % ( ' ( ! ! ' F 7(* G N 5_7 (*'65_7 '
"
. J L ( 1; - ( ! " c a Q j + - lV- 0& 9. & ' % ( ( ( ! ' 2! F (:7G N (' V( 6(*( '6' 5_ 7 5 (*'65_7 ¤ ' & 1 1 ! ,-Q
5 g5_7! " . £) ¢ 5 g5:'g5: O! " a a a 1 Q j + l 0& 10. 5_7! "
& 5 65 5_77! ! ' & F ( G ¤ ' ( Q * ( ' ! " . g5 7! " F E(:7G c Q j + - l 0& 11. & ( ' ( ' ( + ,-Q1 . 5_7! ' 5_7 ( ' % ! ' g( P a ¤ 5_7X! ' F E(:7G N
5_7 (*! '65_ 7 1 5_7 (:7 . (:7G - V-;&-21 Q j + - l & F (:7G -; F 5_7G ¤ F 12. &-
5N 3
.
13.
14.
7 g5_7X! " F E: ( 7 G c a Q j + &( !' ' % g 5_7X! ' F
; !, 3
. - l 0&
( ' (')( V (')( 7<5 7 (8 G & N 1 ( 'g5 7 1 - ! a . +1 5 7 (:7 . (:7G -;& - Q j + - lV- & F (87G -; F 5_7G ¤ F & -
#$
¤
7 ( . £ (9! " ( ( ! " F 7 (8 G c a Q j + - l - &
15. & (' (6( ' 5 (*'65_7 #$ ' ( 9! ' ( ! ' (:7 1 ' % ( ( ! F 7(* ¤ G N '
7 * ( 6 ( O ' * ( ' 5_7 1- &
6 ! 1;- 9! " . ¢ g( 5_ 7X! " c a Q j + - l 0& 16.
3
&-
#$ 5 g5_7 (' 1 ( '
9'! ! %' N " !A M 5_7X! ¤
P 5 g _ 5 7 C 7 (
& 1 ! 1
9! " . ¢ g( 5_ 7X! " c a a Q j + - l 0&
3 17.
&- 1 #$ ( ' ( ' ( 1 5 g5_7 &! ' & a 1 O '6 5_7X! % & N 5 65_7 C7 ( " !A M 5_7X! ¤ 1 ! 1
&! " . c a 0& j + - l V- 0& 18. P 5 P n 5 ' ! " &1L" J
9! ' j + c a -21 l c a -21 . & ¤ 5 1 L ' J
( &! " . ( ( 2! " F (:7 G c a Q j + - l - & 19. & . W #$ ( a a a a ( & ! ( ! W ' ' 19- -P 1 - ! + 19- ¤ ' % ( ( 2! F (*7G N
' & - a 1 - a 7( ( '
5N95
&' 3 465 78
5.12.4. Sums containing products of Pm
(ρ pk, σ qk) (z) nk
1.
2.
3.
4.
5.
6.
. . 5 g5n'65_7X! " 7 (8 65_7X! " G j + l - Q j + a l a & F & ( ' 65 5_7 5 g5:'g5 7 _ 5 X 7 ! ' ' % N& 65_7 5 7 + 1;- . + 1;- . ¤ ! 5 g5n'65_7X! " 7 (8 . a a .
( ( ! " F 7( G j + - l - Q j + l 0& & 65n'65_7 5 7 ! ' ' % N ( ( ' ( ( 5
¤ _ 5 7
& ! 1- 1; ( ' ( 2 ! 7 8 ( " . . 5_7X! " F G j + - l Q j + - l & & ( ' ( ' ( 65 #$ 7! ' 7,5 5_7 5_ ' % F G & N 5_7 5 7 + a 1 . + ,-Q1 . ¤ . !
X+ a 1 5 5 & L " ¢ ' a l a 1 . & j + a &- l &- -21 . 0& J + j
$ 4 3 L S S -; a J 1 ( ( ( ' L " J ( ! ' 5 1 L ' 5_7X! ( 5n 5n O! ' ( ( ! ( " ' 3 ¢ S J % ! 5 5 1 L ' 5_7X! ' ')(* J 5n'65 5n 5:'g5: #$ 1 Y N = > 'I@ B
&
5:'g5 7 5n'65_7 ! 1;-
5 g5 5_7! " (87 a a . a .
+ l & j + -;9l a -; & j 9 $ % S , ) & 5:' L j + -;9l V-; . 0& ¤ SEa J 5 g5 5_7! " (87 . .
+ a l a & j + a a2S l a -; & j 9 $ % S , ) & 5n' L j + l -; . 0& ¤ SEa J
5.12.5. Sums containing Pm
(ρ pk, σ qk) (ϕ(k, nk
1.
¢ ' . j S + l & ¡ $ 4 3 L J
z))
= 'I@CB 5N;
.
2.
3.
4.
5.
6.
7.
8.
9.
; !, ;
¢ ' $ L j + l . & ¢ F3 G 3 ES a J S 5 g5 5n 'g5 7! " a a a a . ¤ Y a $ 5n'! % F ,G j S + - 9l Q ¢ ' S j + l . F $$ 5n (* G 3 ¢ S ¢ S S l $ L 4 3 h 3 & S J 65_7! ( S '8> @CB 1 3 = % . $m 5 ¢ '$ L 4 3 f & SEa j SE+ a l F $ m 5 G J S ( !" . - ¢ 43fI S F3 m G ( $ 5n + l a a F G ¤ ' ! j % a S "$ " ¢ F3 G j + a l a . ¢ & & $ % &5 g5n'65 7! ¤ 1 3 5n O! ']" # ')
(87! % m . $ m 5 ! '])" "" ')( $ ! % ( ( 2! " F 8G j + - l - ¢ & '
' 5 J !A ( L ( ! j + -;9lV-; . F ¢ 3 G ¤ ' m m ] ' " ) " " . $ m 5 7! ')( $ ! % ( ( 2! " F G j + - l - ¢ f ¢ 4& ' ( ! ' ( L ' % ' 5_7X!AJ ( ( 2! ¤ ' m " ( L . ¢ $ " ¢ J $ 5n '! % j + l
& 5 $ m & 1 "m " (')(:7 g5n' #7 3 ] ' % ! 3 '65_7X! m % g5:'! N W 3 ¢ ¤ V ( m m &
!
. ¢ 65n'65_7! " $ 5n'! % F3 ,G j + l O&
& 1 ] 7' % ! N (' g5n'65_7 #7 3 ¢ ¤ &
5N :
&' 3 465 78
10.
11.
65n'65_7X! " $ 5:'! % F 3 G
1 5 ' 5n7X! " 8 $ 5*'! % F]3 , G
1
g 5n'65_7X! . j + l ¢
& 3 ')(87! % ¤ &-
. j + l ¢
& & ' 5n7X! 3 Q58' 5: ]7 !A'g ( 58 ! ' % 58 ! 3 Q 5* 'g(*7X! % ¤
. 65n'65_7X! " $ 5:'! % F3 G j + l ¢
&
&1 65n'65_7!A 65n'65n !A g5n'65 ! ] g5:'g5 7 !A 65n'65n O! 3 # ')( !% 7X ')(* ! % 3 g 5n'65_7X! ¤ ')(87! % 3
12.
13.
14.
15.
16.
17.
)£ ! " 65 m 5n'65_7X! " a . ¢ )£ I -21 9$ % m 5n'65_7! " F]3 G j & + - l I & 5 7X! ' ¤ '_ % a . ¢ £! ¢ £) ¢ 65n'65n O! " n 5 6 ' 5 7 ! $ %
F3 G j &+ - 1Rl & !- & '65n O! ' 7 % # 6V 5n ')(:7X! % ¤ a . ¢ £! ¢ )£ ¢ 65n'65n O! " 5n'65 7! % F 3 ,G j + 1Rl 4
& 4 - $ & 7 # g5n'65n ! !A g5n'65 ' % 7 ')(:7X! % 65n'6 O5n # ')( O! % $ 5_7X! "'& g5n'65n ! " a . ¢ £! ¢ 5n'65_7! % F3 G j + 1Rl & $ 5 7! 5 m $ & " ( ' (:7#7 g5:'g5 # '65_7X! g5:'g5 7! ¢ 3 N W % m & 1- 1 a !
. $ ']" a F¢ $ G & $ % 3 g3 j + l &1 5 65n'65_7! ']" 5n'! 3
')(:7! % F G 5N<
! ¤
7 ¤
&-21 ¤
.
18.
19.
20.
. ( $ m ! " $ m 5 ! #' ")"" ')( $ ! % 5_X7 ! " j + lV- F ¢ $ G '#" ' 1 3 ' % ' 5 !A 5_7! j + l -; . F ¢ 3 m ' m . ( $ m " $ m 5 ! '#)" "" ' ( $ ! % g5 7! " 0 j + - l F 5 $ G ' 5 ' 5 m !A( g ! 5 7! j + -;9l . F m ' m m ( ( 7X! " . $ m 5 ¢ + l V ')( $ ! % 5_7! " j f &-21 F $ m 5_7G 7 (8 5 g5_7X! ' ' % '
m 5_7X!A 5_7X! ' F G
5.12.6. Sums containing Pn functions
(ρ pk, σ qk) (ϕ(k, mk
4 3 V
3.
4 3 V
G ¤
G ¤
¤
z)) and special
1 ( 'L" ' ( J $ ! % 5_7! " &- F ')( $ 5_7 G '65_7! '#" ' % $ G
F '65_7 G 1 (*' L " . J ')( $ 5 7! % 5_7! " j + lV- F ¢ '65_7 G = '*>?7R@CB (*' ¢ &-21 ')( J 1 $ ! % L5_" 7X! " Y F $ 5_7 G j + lV- . F ¢ $ 5_7 G
&-
(')(:7V(')( 5 #$ ! '%& " 1 ¢ ¤ '65_7! % O'65 7!A
5 2! & N 1 3
! W
( 1 (*' L " ¢ &-21 ')J ( $ ! % 5_7! " Y a F $ 5_7 GTj + lV- . F ¢ $ 5_7G
&- 1 (')(:7V(')( 5 #$ '%& " & ! ¢ ¤ '65_7! % O'65 !A
5 2! & N 1 3
W !
¢ 1. 3 &- -21 1 Y j + l V- . F ¢ '65 7 ¢ £ G 3 Q F 2.
; !, :
5N L
4.
$ 5_7! "" 5 7! "
&' 3 465 78
. a a ¢ Q j + l - F ¢ & - ¢ D ! 7 '& . ¢ a ¢ c 4 Q j + l V- F $ & ]! " '6 5_ 7! % 5 ! ¢ 3
$ 5_7 G
7 -2a 1 1 ]F 3 G 3 '65 7! % H ¤ $ 5_7! "" 5 7 ! _ 5 7G 5. " ( ' (87 5 ¤
N ! (
. ¢ $ 5_7! "" a 5 7! " 43 Q c 4 3 Q j + lV- F $ 5_7G 6. & 7X! " F 3 G $ 5_5 7X! g% 5_5_ 7! " c a ¢ Q ¤ & ('6(*'! " . ¢ ¢ 7. &-21 5_7! " c F $ 5 7 GTj + l V- F $ 5_7 G & % ' & " (')(:7V( 6( ' (87
5 43 ¢ '65 7! % ( P5n'65_ ¢ ¤ 7X!A 5 2! & N 1 3 !
. ¢ ¢ &-21 - -; &-21 F $ 5_7Ggj + l - F $ 5_7G 8. &' " '& a ] ( 7X5 ! 1 - a - -;&-21 F]3 !G ¤ 6 ' _ 5 7 ! , G % 3 F 1 ( (*'! " ¢ a . F ¢ $ 5_ 7G &-21 C7(%9! " c - F $ 5_7 GTj + l 9.
& "'#" 65_7! '%& " '%& '65_ 7X! % 5 5
g5_7X! ' 5n'65_7!A( Q(:7X! ( ' (:7# 6(:7V( ( ' (87 W ¤ Y ¢ 3 N ( ( ( ( O')(: 7 ( &
1 !
¢ 5 g5n'65_7! " -21 C7 ( 9! " 10. F3 G Y c - F $ 5_7 GTj + a l a . ¢ ¢
&- ! " ( ')(:7 # Q(:7 5 g5n' _ 5 X 7 ! ' '65_ 7X! % ( Q(:7!A
¢ W 3 &N
¤ 5 g5:'! V( 1 ! (
5N P
.
; !, <
( (*'! " ¢ &-21 C7 (9! " '%& 5 g5_7X! ' & " L '65_7X! % 5 g5_7X! J 5n ' '65_7!
. c - a 1 F $ _ 5 7 GTj &+ - l a F ¢ $ 5_7 G W ( ' 8 ( 7 # ( (*')(:7 ( #$ ¢ ¤ ! ( ( (* ')(:7 1 3 &N
5 g5n'65_7! " ¢ C7 (9! " -21
12. F3 G Y c - a F $ 5_7 GTj + a l a . ¢ ¢
1 &" - (')(:7 #
5 g5n' 7X! ' ! '6 5 75_ W ! % 5 g5n'! & N
3 ¢ ¤ 1 (
!
9! " & -21 ¢ ¢ -21 &- ( 5_ 4 3 3 13. 7X! " Y c a F ')( $ GTj + l V- . F ¢ $ 5_7G ']" ( &&- ! ' 6 ' 5 7 ! ( &! '
5 g5_7X! ' . ¢ 43 ¢ a 1 5_7! j 6 ' _ 5 7 5 7 ! G G + 3 F F l ; ' ' " ( (*'! " '65_7 a Y ')( $ 5_7! % C7 % ( 6(*'! " ( ( )(*'! " F G c &- F 'g5 7 G ¤
11.
14.
& -21 ¢ ¢ -21 &- a 1 ( 9! " 5_7X! " 43 3 Y c a a F ')( $ Ggj + lV- . F ¢ $ 5_7 G
&- 1 ']" £ 43 ¢ a 1 '65_ 7X! 5_7X! (&! '%& j + lV-; . F ¢ '65_ 7G ' 7! ( & !
5 6 _ 5 ' ' ¢ 5_7! ' / F]3 ,G " ( (*'! '65_7 Y ')( $ 5_7! % C7 (%6 (*'! " " ( ( )(*'! " F G c a a &- F 'g5 7 G ¤
1
1.
¢ ' . ¡ $ L 4 3 j S + l &
J
5)N
(ρ pk, σ qk) (ϕ(k, mk
5.12.7. Sums containing products of Pn
z))
=
'I@CB
2.
6 !
65 5n'65_7! " ¢ 6_ 5 7! " -21 F G Y j + a l a . 4 ¢ ¢ j + - l . F $ 5_7 3 ¢ G & ( ' 8 ( 7 65 5:' 5 7! ' ! " '65_ 7X ! % 665_ 5 5:'!A 5 ! & N
( 3 ¢ ¤ !
(*'! " & -21 ¢ ( 6 _ 5 7X! " ¢ -21 3 & 3 3. Y j + l - . F ¢ $ 5 7 G j + l a . F ¢ ) ' ( $ G ']" 5_7! ' & 6 ' _ 5 X 7 ! . 3 g5 7! ' 5 7 G j + l -; F ¢ '6_ 6 5 5 7 ! ' ¢ 65_7! ' F G " ( ( '! '65_7
. Y ')( $ 5_7! % ( )( ( " '! " 5_7X! " F + l a & - F ¢ '65_7 G ¤ G j 5.13. The Legendre Function Pνµ (z) 5.13.1. Sums containing Pνµ 1.
2.
k k (z)
£ ' (65_7! " ¢ j $ L h 3 O
& 4 3 a ( * 6 ' 5 J &L " J £ -; ( )( ! ' ¢ 3h
-; j -; 0& = C7 ( ( 1L' -; J
£ ' ( )( ! " ¢ j $ L
4 3 h 3 & ( * ( ' J 1 " L J £ -; (65_7! ' ¢ 3h
-; j -; & = C7 5 1L ' a J
]!
@ B
]!
@ B
5.14. The Kummer Confluent Hypergeometric Function 1 F1 (a; b; z) 5.14.1. Sums containing 1 F1 (a; b; z) 1.
m !' m m 5:' ¢ ' ( m !" ! " 1 N 1 5 ! $ L ! ' 1 N 1 5n' ! L ¤ $ 4 3 L J J J 5)
.
2.
3.
4.
5.
6.
7.
( m !" ( m #7( m ' ! " C7( (*'! " 3P& 1 N 1 5 ! $ L N #7 ( (*!' $ L J J J m m (*' ' ( ]! " ! " 1 N 1 5 ! $L 1 N 1 L ¤ $ L J J J ! C7 ( ! " ( 5_7! ' m m ( ¢ ' C7 ( (*'! " 1 N 1 ( ! $ L !' $ 4 3 L N
J ( J J C7 ( ! " m ¢ ' ( (*'! " 1 N 1 ( ! $ L $ L 4 3 J J (: 7m ! ! ' ! 3& N m ' ' 1 1J C7( ! " m !' m m 5n' ¢ ' m ( 5 7! " 1 N 1 ( ! $L m ( 5 7! ' 1 N 1 L $ L 4 3 J J J ! m ¢ m ( ' ¤ ' ¢ $ L 3 O - 1 N 1 ( ! $ L 3 O -; 1 N 1 (*' ! L J J J
; 5) ,
¤ L
¤ L
!(*'
5n' ¤ 5n' ! L ¤
5.14.2. Sums containing 1 F1 (a; b; z) and special functions 1.
2.
3.
4.
5.
m! ( m !" a ! 5 $9% " 3& &- ;- &-21 43& 1 N 1 J $ L ( m !" ! 9$ % " ( 7! " ¢ $&% ( 7! " ¢ $&% ( 7! " ¢ $&%
3&
3O
3O
3O
43 ¢ C 7 ( ' % ! ' N m (* ! ' L ¤ 1 1J m ! m !' m 5:' ¤ a 5 ' $ L % -21 3P& N N 1 1J 1 1J ! L & ( m !' m m (*' ' % 1N 1 L¤ -&- & 1 N 1 J ( ! $ L J ! ( m !' m m! ¤ a ' L -&- -; 1 0& 1 N 1 J ( ! $ L ' % ! ' 3P& 1 N 1 J 5n m ( $ a -&- -; 1 43& 1 N 1 J ( $! L ' % m ! ' ! N m 5:5n' ' ! L ¤ ' 1 1J 5)!,
6.
C 7 ( ! " ')( $ ! % m ( 5 7! " 43& - c - 4 3& 1 ' C7 % ( ( ! ' 5 ( m
!
m N 1 J (! $ L &! ' m # 5_7 7! ' -; N (*' Q(*! '65_7 L ¤ J
5.14.3. Sums containing products of 1 F1 (a; b; z) 1.
m ( ¢ O ' ( m ! " C7 ( (* '! " ! " m ( (* '65_7X! " 1 N 1 5n! O')( $ L $ L 4 3 J J 5n' !' ( ! £ m ! m ( ' ! J 1 L ' N m ' m ' & 5:'
m N1 1 J 5 ! $ L W ( m 5n' '65 1 #$ a 5n' 5n ' ! ¤ 1 1
5.15. The Tricomi Confluent Hypergeometric Function Ψ(a; b; z) 5.15.1. Sums containing Ψ(a; b; z) 1.
2.
5 ' ¤ m ¢ m : ¢ ' 5 '! L $ 5 ! $ L 4 3 0 fI n 4 3 L J J J
' ( ! " ( ! J $ L m "
m ( ( 5! $ L J
7! ' m !'
m (*' ¤ J ! L
3.
5 $ ' m 5 $! L $ L 0 I f J J
4.
m m 5n' ¤ ¢ ' $ ( ! $ L 0 fI 4 3 L J J J ! L
5.
m ( ' ! ¤ ' m! ¢ ( ' L * $ ( $ L -; L 0 g f 3 J J J
6.
¢ ' m ( 5_7! " ( * ( '! " 43 J $L
7.
m !" ' m 5 $ ! m ( 5_7! ' ( ' ! " 3P& 5 C 7 ( ! ' $ L J J $ L
J
m! ¤ ' L 5n
m m !' (:X7 ! ' ( ! $ L J
5)3
m 5n' ! ¤ 5n' L J
J
m ¤ (! ' L
.
; !; ,
5.15.2. Sums containing Ψ(a; b; z) and special functions 1.
2.
3.
4.
( ]! " &$ % ( ]! " $&% ( 7! " $&% ( 7! " $&%
m a 5! $ L -21 43& J &m a -&;- -21 43& J 5 ! $ L
0fg3 ¢ & -;- & - a 1 0fg3 ¢ & -- &
m !' 5 ' m n ' % fT3 ¢ L J ! 7X! ' m 43 ¢ m ( ' 5_ % (! J m ! ( ! ' m! ' % ' 5n 0& J ( $ L J m ! ( 7! ' m (*' ¤ ( ' % J ! L J $ L
¤
¤ ' L L¤
5.16. The Gauss Hypergeometric Function 2 F1 (a, b; c; z) 5.16.1. Sums containing 2 F1 (a, b; c; z) 1.
2.
3.
4.
5.
6.
7.
m ! (#! ' m (K5:' ¤ ¢ ' #! " ! 5 ! $ L ' & N J 5n' ( L 43 J $L " N 1 J ! !" m 5 $ ! m !' m 58' ¤ ¢ ' ( m ( ' 5n7X! " N 1 m ( !' N 1 $ L L 4 3 J J J ! L ( m !" ( !" m ¢ ' ! " C7 (*')( m ( 5 ! " N 1 5 ! $ L $ L 4 3 J J ! ! ' 5n' ¤ ' ' m ! 5 ( ! ¢ 3h& N m 5n5n ' L ' m '
1J ! ( ! ' ! : 5 ' ' m " m m m ¢ ! " N 1 5 ! $ L ! ' N 1 5n' L ¤ $ 4 3 L J J J ! m m (*' ¤ ' ( !" ! " F 7 (8 G N 1 5 ! $ L ¢ 3h& -; N 1 $ L J J J ! L m 5 $ 5 $! m 5:' ¤ ' !" !" N 1 5 $ $ L L N
1 J ! L J J m 5 $ 5 $ ! ( !' m 5n' ' !" ! " 63 ¢ N 1 5 $ ! ' N 1 5n' L ¤ $ L L J J J ! 5)5
8.
9.
10.
11.
m !" !" ' !" * ( '! " $ L J C7 ( ¢ ' ( 43 J $ L ' J $ L ' J $L
m 5 $ 5 $! 5 $ N 1J !" m (*'! " N 1 ( ! $ L J ! m ! ' '
! $
m L N 1 J (*' L ¤ !
!' m (:7X! ' 43& N 1 J C7 ( ! " 7(8 m m 5 ( (*'65_7X! " F G N 1 ( ! $ L J C 7 ( ! ' ( ( ! 43& -; N m ' m
1J C 7 ( ! " 7 (8 m ( 5_7! " F G N 1 ( ! $nL J C( 7( 5_! 7X' ! -; N '
5n' n 5 ' 5:' L ¤ !
(*' * ( ' (*' L ¤ !
( ' ¤ m * * ( 1J ' ! L
5.16.2. Sums containing 2 F1 (a, b; c; z) and special functions 1.
2.
3.
4.
. ( m !" ( !" a !" j + -; -21Rl -;&- - a ¢ 3 £ &
& $ % & C7 ( ' % ! ' 0 Q3
L¤
. ( m !" ( !" a !" j + -; -21Rl - ¢ 3 £ & N 1 & $ % & 43 ¢
L¤
. m !" !" ! " 0Q3h
j + -; a A-21Rl a - ¢ 3 £ & N & $ % & 43 ¢ m !" !" ! " 0Q3h
& $ % Y N
1J
. j + -; a A -21Rl ;- a a - ¢ 3 & C7 m 5 $ 5 $! 5 $ L 3 ¢ 5)!;
m! N 1J 5 $ L m ¢ N * (
1J ' ! m! 5 $ L J C7 ( ! ' m (*' ' % N 1 (*' J ! m 5 $ 5 $! 5 $ L 1J C7 ( ! ' m (*' ' % N 1 (*' J !
L¤
£ & ( !' m ( ' (*' ¤ ' % N 1 ( ' L * J !
.
5.
6.
7.
8.
9.
10.
11.
12.
m !" !" ! " 0Q3h
j + - a A-21Rl & $ % & 5 Y N m
1J C7 ( ! " ¢ j + -; & - a 1Rl a 9 $ % 0 Q 3 & C 7 ( ! " ¢ $ 9 % 0 Q 3 C 7 ( ! " ¢ $ 9 % 0 Q 3 C 7 ( ! " ¢ $ 9 % 0 Q 3 ¢ C 7 ( ! " 9$ % j 43 ¢ C 7 ( ! " 9$ % j 43 ¢ C 7 ( ! " 9$ % j 43
; :1 ,
. -; a a - ¢ 3 £ & $ 5 $ ( m !' m (*' ' % N 1 5 $ ! L J ! . ¢ £ m -21 3 & N 1 ( ! $nL J ! ( ! ' m ' m 5n' ' % ! ' 3P& N 1 5n' J ! . m j + -; &- a 1Rl -; a a - ¢ 3 £ & N 1 ( ! $ L J & ( ! ( ! m ' ' m ' % !' N 1 J 5n' ! . m j + -; &- a 1Rl a -21 ¢ 3 £ & N 1 ( ! $ L J & ! ( ! ' m ' m 5n' ' % ! ' 3P& N 1 5n' J ! . m j + - l -; a a - ¢ 3 £ & N 1 ( ! $nL J & ( m !' m (*' ' % N 1 J ! . m ( $ ( $! + a - Rl -;&- - a ¢ 3 £ & N 1 ( $ L J & m ' ! % ' ¢ 3h& N m 5n'
1 J ! . m ( $ ( $! + -; & - a 1Rl - - a -; ¢ 3 £ & N 1 ( $ L J & m' ! % ' ! ! ' 0Q3h
N m 5n5:' ' 5n ' '
1J ! . m ( $ ( $! + -; & - a 1Rl - a -21 ¢ 3 £ & N 1 ( $ L J &5n' m ! ' ' % ( ! ! ' 3P& N m 5n '
1J ' ! 5):
L¤
L¤
L¤
L¤
L¤
L¤
L¤
L¤
13.
C 7( ! " ¢ j + -; &- a 1Rl ;- a a - . ¢ 3 $ & % 0 Q 3 & ( m' ! % ' ! ( '
! $
£ & N
1J !' 43&
m! ( $ L m N 1 J 5n' L ¤ !
5.16.3. Sums containing products of 2 F1 (a, b; c; z) 1.
2.
3.
m !" !" ' $ L m 5 5 1 L)" m 5 ( 'g5 J J
5J $ 5 $ Y N m 5 5 $ 5 !
1 m 1
m !" !" ' $ L m 5 ( 1 L)" m 5 ( ' ( J J
5J $ 5 $ Y N m 5 5 $ ( !
1 m 1
¢ 43 J Y
4.
' J $L m Y
1 L"
m! N 1 m 5 ( 'g5 N & m
1 L"
$ 5 1 m ] m 5 5n m 5 * ( ! '65 1 ¤
m (:7 ! N 1 m 5 ( 'g5 $ ( 1 (:7 m 5 : ( 7 ¤ N m 5n (* 5 ( ' !( 1 m & m
m ! " ! " 1 ( m ( (*' L " ' J
$ L m 5 5 1 L " C7 ( m (*'! " C7 ( (*'! " J
m 5n')( $ 5n')( $ m 5 $ 5 $! ! m 5 5:' ( $ 5 1
N 1 m 5 5 $ 5 1 N 1
' 5 5n' m m ! ! ' ! ! ' m 5n5 !! ' N m 5:5n' ] 5: 5n ! ¤ ' 5 : 5 g ' 5 ' ' ' m m m m &
1
m !" 5 1 L " m ( 5 ! " J m 5 ( '! " F 7 (8 G N 1 m 5 1 5 ! $ m ( m ] m 5 ( 1! 1 ¢
3h& 1 & N m 5n (:7 m 5 ! ( ' ¤
5 $
N 1 m 5 (*'6 5)!<
.
5.
6.
; !<
m !" !" 1 ( m ( ( ' L " ' ¢ J
$ L 0 Q 3 m 5 5 1 L " C7 ( m (*'! " C7 ( (*'! " J J
5 $ 5 $ m 5 1 5 1 ! #$ m Y N !
1 m 5 5 $ 5 1 N 1 m 5 5:' ( $ 5 1
' m 5 5:' m ! ! ' ! ! ' m 5n5 !! ' ¢ 3h& &-21 N m 5n5n' 5n 5: ' 5 !1 ¤ ' m ' ' m m ' m 5 5n6 &
( ! C 7 ( ( * ( ' ! " " m m ' 1 L" ¢ J ( (*
' C7 ( (*'! " m 5 ! " $ L 4 3 m J & L" J
m 5 m 5 1! 1 Y N
!( $
1 m 5 5 $ N 1 m 5 5n') (:7! ' ! ' m 5 ( 1 L ' m ¢ J m ( 1 L ' ! ' m 5n (: 7X! ' 3h& &-21 J
m 5n')(:7 5n' m 5 5n')( 1 ¤ Y N : 5 n 5 ) ' : ( 7 5 n 5 '
m m &
!
5.17. The Generalized Hypergeometric Function p Fq ((ap ); (bq ); z) 5.17.1. Sums containing p Fq ((ap ) mk; (bq ) nk; z) 1.
2.
3.
m ! ¢ ' m !" ! " N a 1 ! ! 5 $ ` $ 4 3 L ^ J (
m ! ( m 5n' m !' ! ' a 1 N a ! 5:' ( ! m ^ ( ( ' & m ! ¢ ' 5 5n'! " ! a N a !" $ L 4 3 1 1 ^ ! 5 $ ` J '! % ( m ! 4 3 ¢ SEa % 5n ! ! ' a 1 N a 1 ^ ! 5n' m ! ( ' ! " '! " ! N a ! 9 $ % 1 ^ ! $ 5_7 ` ' ! O '! % m ' N a !9m 5n!9' 5n' 'g! 5 7 !' 1^ 5)L
` ¤
` ¤
` ¤
4.
5.
6.
m ! ¢ ' C 7 ( ! " m !" N a 1 ! !( $ ` $ L 4 3 J ^ m 5 ! (:7X! ' a N a m ' 1 ^ m ! m 5 $ ¢ ' m !" ! " a 1 N ! ! ` $ 4 3 L ^ J ( ! m ! ' a N a '
1 m ! $ 5 m ¢ ' m !" ! ! 5 ! 43 J $ L " a 1 N a 1 ^ $ ` ( ! m ! ' '
' m !" 7. J $ L 43& ! " ' ( ]! " m ! " 5_7 ! " 8. J $ L $ (:7X! '65_7 7X! m (:7!
m !95 $ m a 1 N ^ !95 $ ` !
m ! m 5 n 5 ')(87 ! m 5 (:7 ! ` ¤ m ! m m ( 5 7 ¤ ! ( (*6 ' 5_! 7 ` ^ m
m ! m ¤ a 1 N a 1 ^ ! 5:! ' ` m ! m (*' ¤ ! ! ` a 1N ^
m !95 $ m a 1 N ^ !95 $ ` !
m m !(:7 a 1 N ^ !(:7 ` 3 !
a 1N ^
m (*')(:7 m !(:7 ¤ !(:7 ` !
' m !" 9. J $L 4 3& ! " 1 ('! " S 5 ! 10. a2S $ % 43 ¢ S + a a 1 .
( m !95 a 1 N ^ !&5 $ ! m !" 43 ! " a
m !" ( a N !" 1 ^ C 7 "( % C7<5n'! ( C7
5)P
$ `
1 N ^
(
'g5 $ & m !95 !&5 $ m ! ( !0! ( S ( m !0! ( Y a N 1 ^
(*'65 $ & m !95 $ ¤ !95 $ `
!
$ `
43 ¢ J $ L ( (*' & m ! ( !( $ 5 m ! ` ¤ !
.
11.
12.
; !<
m ! " ( 'g5 $ & m !&5n $ J( 1L" !95n $ `
% ) 3 , ! " a 1 N 9 $ ^ ! a N a ( ' ( ' ( 1 & m !
1 ( ')(:7 ! ! m !95 $ m 5n $ (:7 m ! " m ! " ! " ! " N a & !95 $ m 5: $ 5n! $ ( m 5_7 ` ^ 1 m #$ m !95_7 a 1 5_7X! a N a ! m 1 !95_7 m 5 7 5_7 a & ( m 5_7
m !' & ' & 3 m ! '%& ! ' & ! '%& #$ m !95n'65_7 a 1 5n' ! Y a N a !95:'g5 7 5n '65_7 5n '6 5_7 a 5n' ( 5_7 m m 1 &
. ' ( g ' : 5 ] ) ' * ( ! $ $ m + & -21 $L a N ! `
J ^ ! ' £ &-21 a N (' 1 & m ! 3 7,5? ( 7X! ' ' @ ` !
^ = ! ' $L ' ')( ( $$ 55 77! a N ( 'g5: $ ] ') (*! $ 5n m ! `
^ TJ ! £ a N ( ' & m ! ! O
! ! ' $
1^ ! m ! 7 ')( $ ! % $ 5n'65_7! % N a ! 1 ( $! & 5 $
' m ! '6 5_7X! % N a !
! '65 & 1
¤
13.
14.
15.
16.
5,9N
¤
¤
¤
¤
¤
17.
18.
19.
m !95 $ ( ! " m ! " $ ! % ! " N a 1 !95 $ $! 5n ` ^ ' & m ! '%& ! ¢ 3 O ('Q ]5n O! % ! ' & a 1 N a
m !95 $ " m !" $ 5 7! % ! " N a 1 !95 $ ] $! 5 ^ m !' & ' %& 3 O'65 ! % ! ' &
^
m !95n'65_7'65_7 !95n'65_7'65n '6! 5
` ¤
m ! N a 1 ^ ! ! ] ` m !95n'65_7 N a 1 ^ !95n'65_7 O'6! 5 ` ¤
`
7 O ' 7 m !" G G J $ L F F]3 43
! "
m ( O'65n $ $ ( m !95n $ 1 ¢ ¤ Y a N !95n $
!
20.
( $ & m ! ¢ ' m 5 $ ! ']" ( P5 $ ! " ! a N a m 5 $ ! " $ 4 3 L 1 1 ^ ! #P5 $ ` J ' m ! 5n ' m 6(%5n9!')' (:7 a N a ! (#Q ( m (*! 6 ' 5_7 ` ¤ m 1 1^
5.17.2. Sums containing p Fq ((ap ) mk; (bq ) nk; z) and special functions 1.
2.
( 7! " ' ¢ $ & % $ L I f J C7( ' % m ! '
( $ & m ! a 1 N ^ ! ` ! ¢ a N a ( ' & ! m ! (*m ' ! `
1^ m ¢ & -21 $& %m ' (*( '! $ " ! a N a ( $ ! m ! ( 3
1^ m ('65 $ & m !95 $ ' ( ! " m ! " !95 $ & ! J $L 1 L" ! " a 1 N a 1 J
a N a (' m ! 1 1 5,
m! ¤ ' `
! ¤ ! 1
.
; !< 3
m !" ( 'g5 $ & m &! 5 $ # 7 ' !95 $ ] ! ` $ L 4 3 & Q ! " a N a 3. 1 1^ J N ( ' ! m ! ¤ ^ ! ` 5.17.3. Sums containing p Fq ((ap ) mk; (bq ) nk; ϕ(k, z)) ( & m ! ¡ ¢ ' 'I@CB a N ! 5 $ L $ 4 3 = 1. ` 1 ^ J ! 2.
m m !" ' ( g ' 5 9 ! 5 $ $ m 2- 1 ¤ 9 ! 5 $ $ $ L a N ` ! " 1 ^ J ! 1 ( '65 $ m !95 $ $ $ "" m !" !&5 $ $ 5_! 7 ` ')( $ ! % $ ! % ! " a 1 N a 1 ^ 1 m m 5_7X! m 3 ')(* ! % 5 7! ')(:7! %
3.
= '8> X@ B
(' 5 $ I m !5 $ $ 5 7X! $ 5 7! "" m ! " '*( $ ! % $ 5 7 ! % ! " a 1 N !25 $ # ! $ h 5 ` ^
4.
m ' % 3 # '*(?7! % ¤
7
5.
( 'g5 $ m 9! 5 $ ' $ m 5 !" m !" 5m _7 ! " a 1 N !&5 $ $ m _ 5 7X! ` $ $ L J ^ !
6.
( ¢ ' S a N $ L 4 3 1 ^ J 1
( ' & m ! 1 C7( ! #$ ¤ ! a N a 1 ! 1 5_7
m ! ¢ ? S l ! ! ` 3 S m !( 3 ! ( 3P& S
5, ,
=
'I@ B
7.
8.
9.
¢ ' S a N 1 4 3 J $ L 1 ¢ S S 3 -;
' ( $ V m ! ! f O &- -21 a 1 N J $L 43 fI ! 1 ' 3 &-21 ' 5 a N ( ' ! V ( m ! ¤ m 1 !
( $ m ! ' ¢ -21 &- -21 a 1 N ! J $L fg3 ! a 1 ' ( ' 5 7 ! m m '65_7!& (*'! a N ! m 1 ! a (:7! m 5_7! ' ¢ 3 a '65_7! m (:7X!
11.
12.
m !( m ! ¢ E S a S ! ` 3 1 ! ( ! ^ m ! ( "' S -; ¢ a a 1. ! ( " ' a J $ 5n' L 43 + ' ( ( 5_7! Y ' ( ( 5 7! " = > 'I@ B " m
(
10.
! 1
( ' m ! (:7 ¤ 1 N !(87 ! a 1
( $ & m ! ' m !' ¤ ¢ ' ¢ f &-21 a 1 N ! ' m 5_7 ! ' 43 J $L ! a 1 ( m ! #$ ¢ ' ; 1 ¢ f 4 &-21 a N ! W 43 J $*L ! a . + 1 ! ' J 1 L 5_' 7 m ' ¤ !' m ( m ! ; 1
a 1 ¢ O '65_7 ¢ f 4 a N ! W 3 J $ L ! a . + 1 5,93 #$
¡\¤
.
; !< 5
5.17.4. Sums containing p Fq ((ap ) mk; (bq ) nk; ϕ(k, z)) and special functions 1.
( $ & m ! J 1 (*' L " ¢
')( $ ! % 3# &-21 &- F $ 5_7 G a 1 N ! $&% ! a 1 (:7X! '%& ' & " (')(:7V(')( 1 m ! (:7 3 '65_7! % O'65 7! ¢ ¤ " a N 3 ! : ( 7 (
m (87!
!
2.
J ( ')( 1 L " ' ( $ ! % 4 3V ¢ &-21
& $ % ( $ ! Y a F $ 5_7 G a N ! m
&- 1 1 ! a 1 (:7X! '%& ' & " (')(:7 V(')( & m !(:7 3 '65_7! % O'65 ! ¢ ¤ " a 3 N ! : ( 7 (
m (87!
!
5 7! '#" $ _ &$ % 3 d 3 3. '& " ' 5: ( 7! % ! ( E5* ' 5n7! m
$ 5_7! ""
4.
5.
4 3 1
$&%
( $ & m ! c&- F $ 5_7G a 1 N ! ! a 1 ( 7! ('g(*7]( ( 'g(*7# m !I( 7 a N !I( 7 ( " ` 3 ¢ ¤
^ (*7X! !
( $ & m ! 43 Q 1 N ! ! a ! " m ! 1 " a ¢ ¤ $ 5_7! % ! " c 4 Q & ( $ V m ! ( ( '! " ¢ $&% &c - F 'g( $ 5n7 G a 1 N ! 3 &- -21 ! 3Q ¢ &-21 c F '65_7 G ( $ & m ! ( ! ' '65_7 '65_7 '65_7X! % G a 1 N ! ¤ J $ L 43 3 d2 F ! a 1
c&a - 4 3 Q a
5,!5
5.17.5. Sums containing products of p Fq ((ap ) mk; (bq ) nk; ϕ(k, z)) 1.
2.
('65 $ m ( $ & m ! ' 5_7 ¢ 6 4 3 J $ L N 1 ! '% & a 1 a ' 1 N ! ! 1 5 7! m ! ' & ( 7X! ( '6 _ 5 7!A ! ' m ( 5_7 (')(:7 m ! #7( (*' Y m '65 m 3 a N a
1 ! V( m (*' ! a (:7X!A ! ' 1 ( ' (:7 m m ! ' & a 1 N ! ! a ( ! $ ' m & -21 ¢ -21 3 &- a N ! a $ L ! 1 1 EJ ( ' & m ! ('65 $ & ! ¢ Y a N ! & 2 1 3Q a 1 N ! 1 ! a ! ( ! ' Em - ! ' a 1 '65_7 ¢ a a .M '65_7 1 G '65_7 ! ' 1 F KJ $ L 43 A+ C7 ( (*'! " ( $ & Y C7 ( (*'! a N ! " 1 m !
! ¤ 1
5.17.6. Various sums containing p Fq ((ap )+mk; (bq )+nk; z) 1.
!" ¢ ' !" $ L 4 3 J
2.
Y
3.
!
a 1
¤
( !" m 5 $ (K5_7! " & N 5 $ 7 L J ! C7 (#! ' m (*' (K5_7X! ' & N P(*' 7 L = K5 ( m ( ! @ B J ! ' £ £) m ! " m 5n'! " J $L - m 5 1 L " m ! " 1 L " ( 'g5 $ $ J 5 7 $ 5 m $ 5nJ '6 5n m ' % j N & $ 5 & $ 5 m 5 1 $ 5n m 7 0& D m ! ' 0& H
!
= = @ C7 !C@CB ('65 $ $ 5_7 $ 5_7 $ 5:'65 m $9% ! m 5n'! " ')( $ ! % $ ! % 1 L " m 5 1 L " N & $ 5 1 m 5 $ 5 1 ] $ 5n ! 7
J J
'! Y j & 0 ' % m ! % ' 0& ¤
5, ;
.
; L1
$&% ! m ! " m 5n'! " 4. ')( $ ! % $ ! % 1 L " m 5 1 L " m ! " ('65 $ $ J 5_ 7 $ J 5_7 $ 5 m $ 5n'65n m ' %! & 0 ¤ Y N
j & 0 5 5 5 n 5 ] n 5 7 $ $ $ $ m m ' 1 m 1 !
( 'g5 $ $ 5 $ 5 $ 5n'65n #$ ' ! : 5 ' !" m m " & m m £ $ L 5.
N
$ 5: $ 5 m 5 1 $ 5n m 7 m 5 1 m ! " & J ! J L" ' %
Y 0& D m ! ' 0& H = = @ C7 !C@CB
n 5 ' ! ( 'g5 $ $ 5 & $ 5n $ 5:'g5 m #$ " & L " m J
6. $ 5_7 m 5 $ 5 1 $ 5 7 ')( $ ! % $ 5_7X! % m 5 1 L " N & J
!'! % Y & 0 ' % m ! ' 0& ¤
('65 $ $ 5 $ 5n $ 5 $ 5:'g5: #$ " " m m ' m ! " m 5:'! " & $ L 7. N $ 5_7 m 5 $ 5 1 $ 5n m ] $ 5 7 m 5 1 m ! " & J ! J L"
' % Y & 0 D m ! ' 0& H ¤ 5.18. Multiple Sums 5.18.1. Sums containing Bessel functions
; $ # $ ? ¡\¢!!£n¤4¤¤
S 7 & 1. <1
U a a WC[ $ % ZZZ 1 ' '#" ( & 7! ( " S ¢ ' % ]! 1 E S a 7
1 0& 2. <1
U a a WC[ U $ % ZZZ 1 ' ' % 7! ! ( S ( 5_$ 7 J 1 " S J % $ 3. -21
U a a X[, ZZZ 1
J
( $ L a 1 &-21 !£ & = '*>?7R@CB
' >?7R@CB L a 1 <1 £) ¢ && = * 3 ( " 8 M 5 B B B;5 ( ! '%& L
' % M BB BA ( ! Y 1 ¤¤4¤ S ¤
& 2 1 5,9:
5.18.2. Sums containing orthogonal polynomials
; $ # $ ? ¡\¢!!£n¤4¤¤ S j 0& S 0& ¤ 1. Ua a [ ZZZ 1 S S & ¤ 0& 2. U a a X[, ZZZ 1 S 7 0 3# S -21 ¤4¤¤ ¤ $ % 3. S
1 U a a X[, ZZZ 1 S 7 4. U a a [ $ % a 1 ZZZ 1 3 ¢ £ a2S 0 ¤4¤¤ S 1 & S -21 0 ¤¤4¤ ¤ S 1 1 S U a a [ a2S 0 c 5. -21 0 1 ¤¤4¤ S ¤ c ZZZ c U a a X[, ZZZ 1 " S 6. U a a [ $ % - ZZZ 1 5hB B B5 ( ! % S " ' % $ % &- &- = #7 &B BB ' ! 1 5hB B B5 ( ! 5 B B B;5 ( @ B S U [ c 0& c a Z Z Z a c & ¤ 7. U a a X[, ZZZ 1 S ( ! " (* ! " c & $ 8. Ua a [ ZZZ 1 ') (*(' (_(_B B B BB B((* ( ! ( ' ! c U a Z Z Z a c [ & ¤ ' S . + - l - 0& j + -;9l -; . 0& j 9. Ua a [ ZZZ 1 = 5hB B B5 ( ! 5 B BB5 ( @CB
5, <
Chapter 6
Infinite Series 6.1. Elementary Functions 6.1.1. Series containing algebraic functions 1.
"'( & ( m !" 5 ! $ % ! " " ( C 7 ( !0! ( S m !( ¢ 3 ¢ S + a a 1 . % C7 ( m !0! ( 4 3 J $ L N ^ !(
!
$ =
`
@ B
6.1.2. Series containing the exponential function
¢ 3 =
7 7 $&$ K $ ! ' (:7 3 7 3
4-4# $!
' 1 ( ' ' 1 ' ' 1 O'65
1.
2.
3.
4.
5.
6.
K
= = @& 7]B !C@CB
$ !
7 5 _7
= = @& 7]B !C@CB
7
$
K
7 7 7! 3 ' & 8 5_7
( 7X! ' 7 ' 5 7 ' _ 1
= = @& 7]B 7X!C@CB
7 $ 3 7 3 7 7 $ (:7
7X! '
7 7 3 ' & 8 (:7 3 O 6 ' 5 7 !
K
7( $ $ 5,9P
K
$ !
$ K $ !
= = @& 7]B O!C@CB
= = @& 7]B !C@CB
= = @& 7]B !C@CB
:1 3
6.1.3. Series containing hyperbolic functions
4-4# $! 1.
3.
4.
5.
6.
7.
9.
10.
= = @& 7]B 7X!C@CB
1 ' !(:7 7 $ $ K $ ! ¢ ' 3 3 3 1 $ ' ! 7 ' 7 3 $ 1 ' ! ¢ 7X 7 $& $ ' 3 1 $ $ ! 7( ' ! ' 3
K 1 7( ' ! 7 ( $ ¢ ' $ 3 1 ( 7X! ' £ ¢ 3 7 O 6 ' _ 5 7
2.
8.
K ¢ 3 =
' !(:7 7 $ $ K $ ! ' 3 3 K
( 7X! ' O'65_7
¢ 3 1
'65_7X! 7
= = @& 7]B !C@CB
= = @& 7]B !C@CB
= = @& 7]B !C@CB
= = @& 7]B !C@CB
= = @& 7]B O!C@CB
= = @ 7]B 7 !C@MB
( 7 $ :
= = @& 7]B !C@CB
7 3 7X
K
( 7X! ' O'65 7! ( &
6 ' _ 5 7 ! ( & !& ( " & S ( 7X! 5 7! % * ( &! % ES a 1 _
= = @ 7]B 7 !C@MB
53 N
P5_7X!
= @ B
6.1.4. Series containing trigonometric functions
( " S - "'& ! ¢ S 3 ] (:7X! % $ 5_7X! % "" S - -21 ¢ ¢ $ ( ! % 4 3 43 ! % £ 3 £ 3 £! 3 £ = ( ! ( ! = @ @CB ( " S - ¢ "'& $ ! ( " $ ! ¢ ¢ S 43 # (:7! % 3 3 $ 5 7! % $ 43 1 S Y - -21 3 ¢ ! % ¢ 3 £ a - SEa & £ 3 £ 3 £) 3 £ = ( ! ( ( ! = @ @CB $ 5 " $ 5 5 5 m ( X 7 ! 1 L 1 m L J J
m 5 $ 5 1 L ( $ 5 1L 5 m J
J
= = @ C7B 7 O!C@CB ( 7! " ( $ $ 5 m F f G 7 m m f <3 m 0f 1 = ( @CB ( 7 ! " ( @CB F
f G m f <3 7 f = $ 1
$ ! $ 1. $ ( " 1 ( " Y 4 3 ¢ S -21 ] (
2.
3.
4.
5.
6.2. The Psi Function ψ(z) 6.2.1. Series containing ψ(ka + b) 1.
2.
3.
(87 C
1 " $
C
1 " 7 $ 1
¢ 3
¢ 3
7
¢ 3 < 3
¢ 3 C
531
=
7A@CB
=
7A@CB
¢ 3 ¢ 3 3 & ¢ 3 =
7A@ B
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
7 $ 1 7 $ 5_7X! $ 1 7 $ 5 m !A $ 1
:1 ,
<3 C ¤
¢ 3 5 ! ]
" $ (:7X! $
7 3
= = @ !C@CB
C
7 ¢ ¢ ¢ m ( ! ' <3 f <3
3 C ¢ 3 3 ¢ 3 ¢ ¢ 3 3 ¢ 3 3 ¢ 3
"'& 7 $ 5_7X! 3 ¢ 3 ¢ 3 $ 1 ¢ ¢ ¢ 3 C 3 3 7 7 < 3
$ 1 C <3 3 7 ¤ 3 C $ 1 ( 7 ! " £ 7 £ 3 3 7 7X $ 1 7 £ 7 $ $ 5_7! ¢ 3 , C <3 ¢ ) 1 7 £ 3 $ $ 5_7X! ) 3 , C 3 1 7 C 3 £ ] ¤ $ 3 1 53,
C
0f ¢ ¤
=
=
7A@ B 7R@CB
S -21 7 ¤ $ 1
£ F 7 G ¤
= = @& C7 !C@MB = = @& C7 !C@MB
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
7 $ 1 7 '& $ 1
7
"
7 3 3
' 5n 6
7 '%& $ 5 m ! 1 ( 7X! '%& '65 7! % 1
$&% £
C
<3
] ¤
C
£ 7 3 ¢ 3 £ ¤
7 ' . . ¢ . ¢ ¤ + a f ¢ <3 $ L + & - + a 1 0f f J C
"
7 $9% ! ¢
¡ ¡ 3
£/
3
¤
£ ¤ /
" $ 5_7X! % ¢ 9 $ % 7 £ 3 £ / / £ / E3 £ / £ / ¤ 1 1 " 7 G C £ £ ¢ 3 7 7<5 7A@CB $ F = 7 ( 1 ( 7! " 7 $ F G 3 £ C £ £ ¤ 1 £ C 7 7 G <3
3 = = @& C7O!C@MB $ F 1 £ C £ ( 7 ! " 7 @& C7 !C@MB 7X 3 G F G = = $ 1 7 7 £ £ ¢ 5_7X! F G 3 R C £ £ ¤ $ $ 1 7 7 £ £ 5_7X!A $ 5_7! F G 3 3 C £ 3 £ ¤ $ 53 3
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
:1 ,
7 7 $ 5n !9 $ 5_7X! F G 7 7 3 £¡ £ 3 £ £ 3 7 7 £ £ 2G
3 C $ (87! F 1 7 7 7 F 2G 3 , C 0 ¤ 5 7 ! $ 7 7 $ 5h !; $ 5 7!; $ 5 ! F G 7 7 ¢ g3 £ £ £ 7 #7 ( " 7 £ £ £ G 3? C N & ]! $&% F
7 7 7 G C ,
<3 £ C £ £ 3
9 $ % $ F 1 J1L" 7 7 G , 3 £ C £ £ 0 ¤
$9% $ F 1 7 J1L" 7 m ! " $ F G C £ £ D FOfT3 G 3 fI H 1 7 1L" $ 5_7XJ ! % $ 5_7X! F G 9 £ 3 ¢ C & £ 3 7 1L" $ 5n J ! % $ 5n ! F G 3 ¢ £ £ C ¢ 7 7 1L" $ 5n J ! % $ 5 ! F G F :3 G C 3 7 1L" $ 5 J ! % $ 5 ! F G F 7 3 G C 5395
£
£ ¢
¤
C
£ ¤
¢ £ C £ 3 ¢ ¤
¤ £ 3 ¢)¤
7 F fg3 2 G = m h7 @CB
£ £ £ < 3 ¤
£ 3 4£ £ 3 ¢
¢ 3 ¢
£ ¤
£ 7 7 ¤ 7 3
¤
39.
7 1 J $ L 5_" 7X! F G 3 ¢L£ 9 $ %
40.
1L" 7 G ¢ 3 J
F 5 7X! $ 5_7X! % $ _
41.
7 1L" $ 5_7X! % J $ 5_7X!A $ 5 ! F G F £ 3 G
42.
43.
44.
45.
46.
47.
1L" $ 5n !A J $ 5_7X!A $ 5 ! 9 $ % 7
C
£ ¤
£ £
3 C 3 7X & ¢ 3 £ ¤
7 F G ¢
3 £ ¢
& £ 3 ¢ ¤
C
C
£ 3 ¢ ¡ £ ¤
7 1L" $ 5_7X! % $ 5 J !A $ 5 !A $ 5 ! F G O 3 ¢ 83 " 7 9$ % 1 L " F J 7 £ / 1L" J 5_7X! % F $9% $
"
1 J 1 L "
£¤
G
£
£ /
/ < 3 £
7
G £ 3
C3
£ /
/ ¤
£ ¤
7 7 $ 5 m !A $ 5 m 5 7 ! F G 1 5_7! 7 5_7 ! £ f2 £ m m m
C3
C
£ £ 0 5_7 fI ¤ m
7 F G
7
£ £ £ L / E3 / '
53;
¡!¡
¤
48.
" 5 7 ! % $ 7 D
49.
50.
51.
52.
53.
55.
56.
57.
7
7 F G
J( 1L" J1L"
5_7X ! % $&% $ 1L" J $ 5_7X! % F 9 $ % 1 J 5_ L 7" ! % @ F $ =
54.
:1 ,
/
3 £
/
/
3 7 F G
¢ 3
£ ¤
7
£ 3 G
C3
7
:3 V G
1 " & L " J $ L 5_ J X7 ! % @ F = 1L" $ 5_7XJ ! % $ 5n ! %
C3
7
7 F G
83
7 1L" &L" $ J 5_ X7 ! % J $ 5n ! % F G P3 J 1 L " 7 7 G 7X 83
$9% ! F 1L" 7 J G
$9% = $ 5_7! % @ F 7 D :3 £ 3 C 7 1 L J 5_ 7" ! % @ F G & £ 3 $ =
C
C3
C3
53 :
£/
£ 0 ¤
¤
£ ¤
£
¡ ¤
¢ £ 3 ¢ ¢ £ £ 3 ¢ ¢ 0 ¤ 7 ¤ £
F G
3
7 F G
/
£ 3 ¢ <3
C
£ 3 ; £ 3 ¢ ¤
C
£
£/
£ £ 3 ¢ 9 3
C
G £ 3
£ £
C
£
C
3 ] F G9H ¤
£ :3 ¢ ¡ F G ¤
58.
59.
60.
61.
62.
63.
64.
65.
7 1 J % L $ " 5n O! % @ F G & £ 3 C 3 £ _ 5 X 7 ! $ = ££ ¢ 4¡ £ 3 :3 C 3 3 P3 :3
£
7 F G £!¢ C 3 £
£ F G ¤
&$ % m ! " $ 5_7! fI fg3 ¢ fT3 ¢ < 3 ¢ fT3 ¢ 3 fg3 ¢ fg3 ¢ = m 7 ( " 7 #7 ( " N m 5_! 7 L = = @ C7B 7 $&% fI m f 0fI J 1 m !" ¢ 3 - fI 3 ¢ 3 & $ % fI = m !" $ 5 ! fI 9 $ % ¢ ¢ - 3 1A- 3 7 7 ( ¢ 7( m 3 N 1 ( J " ! m " fI m 1
m 5 7 L fI N 1 J _ ! -21 ¢ 3hf O K3hf 7 7 m #7( ( L 3 C7( m ! & N m! 7 J
7
7A@ B m !C@CB 7A@CB
¢ 3 ( m V7 ( m
¢ h 3 fI ( m V7 ( ( m 7( L ¤ !
mm ( D f - 0 fI 0f N J m 5_7 ! m 5_7 L H = = @ C7]B 7 !C@CB m m (:7 ")" $&% m ! " fI 0fg3 ¢ fI 3 ¢ <D F G G9H 3 F m (:7 m (:7 m D F G 3 F GIH ¤ 7 5_7 m )" " $9% m m 7 D F m 5_ G G 9 G H I f 3 F D F 3 F GIH ! $ " m 1 7 m 5_7 m fI fI <3 D F G 3 F G9H 7 m m 5_7 ¤ G GIH D F F
53<
:1 ,
$ 5n'! % ! 9$ % m " fI fI N 'g5 7 L 3 1A- 1A- 43 fT3 ¢) 1 1J m! 7 3 $ &-- a 1 3 --21 -21 fg3 ¢ 1 ( ! " 5 (:7 5 (:7 Y $9% 5 $ (:7X! 3 N m m 5 $ $ m 5 m $ $ ( L ¤
J & m ! m !" J 1 L " 5_ 7X! % fI 67. $9% $ & ( mL 7 £ 7R@CB J ( m ! D 7( m 0f fI 3 F 3hf G9H = m m !" 1 L " 5_7X! % J $ 5n ! % fI 68. $ ( m L (: ¢ ¢ J m 7 fT3 <3 ( m ! D 3hfI 3 F 3hf G fg3 H = m 7R@CB ( & L m m ! " m J ( fI 3 69. ! 9 $ % J Lm m £ Y 3 0fI 3 D m F m G 3 F m G9H = m 7R@CB m ! " £) ¢ 7 f 70. 4 3 fI $&% ! fI 7 mmm m 5_7 m 5_7 m 5_7 £ ( 7L DMf fI 3 H & N 7 #7 ( 7 L 3 f & fI & N ] J ! J ! = m 7 @CB 7 £! C ¤ 3 71. $ 1 7 7 $ 5_7X! £) 3 C £ £ $ = = @ @CB 72. 1 66.
53 L
73.
74.
$ 1 ( $
7 ) £ ¢ ) 3 , C 3 £ £ 3 ¤ $ 5_7! 7 7X! " a £) ¢ 5 7 1 38D C ¢
H
=
7 7 ,7 5 7 ¢ 8 7 ( H
D
3 < 3
C ( 7
=
7
75.
£) ¢
76.
7 £) ¢ ¢ 43 3 7<5 D C
77.
78.
79.
80.
81.
7
( 7! " " £) ¢ £ £ £ <3 ! @ $ % = ( 7X! " " £! £ 7 £ £ = $ 5_7X! % @
" !£ & $ % fI
fI m
7 #7 ( !
N 5_7
m 5_7
53 P
7A@CB
¢ H
J 1 L " !£ 7 ! G $ % F " ¢ 3 ¢ 3 G 1 7<7 5( 3 F ¢ ¢ F 7(
Y £ C £ ¢ "'& 5 7! % F G a F £) G $
1 " ¢F ¢ 3 G 1 7<7 5( 3 F ¢ 3 ¢ 7(
Y £ C £ ¢
7A@CB
=
7R@CB
@CB
=
£
@CB
=
3 G 1
3 0
=
7R@CB
=
7R@CB
3 G 1
3 0
7 # 7 ( ¤ !
N a &
82.
83.
84.
85.
86.
87.
" 7 $ ! % 0fI £) fI 3 \- \ - 7 7 \- - fI = "'& 7 $ 5 7! % 0fI £! fI ] C7 ( m ! 1A\ - \ 3 1A- 7 7 7 ] C7( m ! D 7( m 0fI H 1A- 3 1A\ - = ( 7! " " $ ! % 0fI £! fI ¢
- R
7 Y f < D 0fI 3 ¢ O H 3 f < = "'& 5 7 ! % 0fI a 1 $ 7 - 3 ( 7! " "'& 5_7X! % 0fI a $ 7 ¢
-R £
\-
£! ¢ 1 f
0f
7 $ 5 7!& $ 5n O! ¢
89.
m !" ! $ % fI
- R
-
m !" $ ! % If 7 £ m DXF - £ G
= 7
\ \- 3 - 0fI
7 & (:7! " ¢ 5_7 3 3 $
¢ 4 3
!
!
7R@CB
7R@B
!
7R@CB
!
7R@CB
£! f ¢
88.
90.
:1 ,
£ 3
0f 2 £ fI 3
7
£ 3 £
¢ 0 =
/
!
£¤
&3
m
= m £ £ fI <3 - 3 5 5N
7A@CB
¤
C 3
m £ fI <3
m H
= m
7R@CB
7R@CB
91.
92.
93.
m !" $ 5 7! % fI " !& 5 7! 7 3 ( 8 m _ D 7( m 3 m
£ fI H 3
m 5_7X!
= m
!" ¢ $ m 5_ 7X! % If 4 3 7: £ £ £ £ m ( 7X! 1A- D -
7 H Dm : ( 7 3 fI H m 5 7 ! = m
m 5_7X!
" m !" 5:O ! % If 3 m (87!A m (* O! $ m 5: O! m ( Y D m (:X7 !A m ( O! 3 £ fI H
m 5n !
7 C 7 F ' & -21 $ O' , < 3 94.
F ' G $ ) 1 1 C ( 7 ! " + ' & -21 $ 3 ' , 7 95.
F $ ) 1 1 $ 5 7 $ ¤ 7 £ G 3 F G9H 96. $ D F 1 $ 5_7 $ 7 £ 7 G 3 F GIH 3
97. $ D F 1 $ 5 7 $ £ £ ( 7 ! " G 3 F G9H 3 98. $ D F 1 7 7 G 3 F GIH G 3 £¤ 99. $ D F 1 $ $ 7 7 £¤ G GIH $ F 3 F 100. G3 D 1
X@ B
@CB
= m
5 5)
7R@CB
= = @ O!C@CB 5_7 ' G = = @ !C@CB
= = @& 7!C@MB = = @& O!C@MB
101.
102.
103.
104.
105.
106.
107.
108.
109.
110.
111.
112.
:1 ,
7 $ $ 7 $ D F G 3 F 9G H ¢L£
1 1
7 $ $ 7 $ D F G 3 F IG H 1
7 $ 5 $ D F
G 3 F
$ 5_ 7
G9H
1
7
1
$ D F
$
G 3 F
$
7
GIH
7
1
$ $ D F 7 G 3 1
7
$
$ D F 7
£¤
G3
G3
7X $ $ 7 7 G 3 F 7 9G H $ F 7 D
1
G3
7 $ $ 7 $ D F G 3 F 9G H 1
£ 3 £ £ /
G
7 $ $ 7 $ D F G 3 F IG H £ 1
£ ¢ /
£¤
£ ¢ / £ ¤
¤
¤
G3
G3
7 $ $ 7 $ D F G 3 F IG H
G3
£ 3 ¢ / £ £
/
¤
¤
£ £ £ / ¢ ¤ 3
/ £ $ 7 ¤ 7F GIH G £ X
/
$
G 3 F 7
7
G9H
G
G3
£ 3 £ /
£/ £
3 / £ " £ / £ ¤
$ $ 7 ( 7X! " ' 5n G 3 F '65n GIH O '65_G7 3 £ £ $ F 6 D
1 $ _ 5 7 P 5 7 S -21 ¢ £ ¤ 3 3 F '65 G " ( 7X! G 7 GIH G 3 / £ ¤ $ 3 F D F 1 5 5,
113.
114.
115.
116.
117.
118.
119.
120.
121.
122.
123.
1
( X7 ! " $ $ 7 $ D F G 3 F 9G H ¢L£
1
1
( X7 ! " $ 5 $ D F
G 3 F
5 7 $ _
G9H 3
£ ¤
G3
£¤
( X7 ! " $ $ 7 $ D F G 3 F 9G H
G3
£
3
$ $ 7 ( 7X! " G 3 F IG H G £ $ F D
1 ¢ £ £ / K3 $ $ 7 ( 7X! " £ G GIH $ 3 F G3 D F 1 $ $ 7 £ £ ( 7X! " G GIH $ 3 F
3 D F G3 1 $ $ 7 ( 7X! " G G9H $ D F 3 F
G3 1 ¢ £ / K3 $ $ 7 ( 7X! " G 3 F 7 IG H $ F 7 D
1
G
7X $ $ 7 ( 7X! " G 3 F 7 9G H $ F 7 D
1
G3
£
¢ /
¤
¤
¢ / £ ¤
£ /
¢ /
£ ¢ /
3 £ 7
£ /
¤
¤
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£ $ $ 7 ( 7X! " ¤ G GIH £ 3 $ F 7 D 3 F 7X
/
1 $ $ 7 £ ( 7X! " G 9G H $ D F 7 3 F 7 G
1 £ £ £ £ ¢/ £ ¤ / "
/ E3 , ) 5 53
( 7! " 5h ¢ $ , 3 )
1
124.
125.
126.
127.
)
$ ¢ ( 7X! " $ F D 3 F 5 G
1 ¢ 3 / ( ¢ ( 7X! " $ , 3 ) ) 1 3 / 5 ( 7X! " $ )
1 / ¢ 3 £
:1 ,
/
5h 7 , £ ¢ 3 3
$ 7 5 9G H £ £ ¢
/ ( 7 , £ 7 £ / £ 7 ¢ 5 7 7 , 3 , ) 7<5 3 £ £ / / E3 £ ) ,
( 7 ¢ ( 7 7 ( 7X! " $ 128. , 3 ) , )
1 £ 3 / ¢ £ £ / / ' £ 7,5 , )
( 7X! " 5 $ 129. )
1 7 , 3 £ 3 *) ( 7X! " ( $ 130. ) 1 3 7 , £ ) *
131.
,
£ /
¢¤
K3 £
£¤
¢E £¤ 3
£ /
£ /
3 £
¤
£¤
¢ 3 5 7 , , ) ¢ £ £ £ £¤
/ K3 / / E3
¢ 3 ( 7 , , ) ¢ £ £ £ £¤
/ ' / / E3
5 ( 7X! " 5 ¢ 7 $ , 3 , ) )
1 / 3 3 £ £ / / K3 £ ,7 5
5 5 5
/
¤
( 7X! " ( $ 132. )
1 3 / £ £ 133.
¢ 3 K( 7 , , ) £ 7<5
/ /
7 5 ( 7X! " 5 ¢ $ , 3 , ) )
1 / 3 3 £ £ / K3 £ 5 7
/ K3 £
£ /
7 ( ( 7X! " ( ¢ $ , 3 , 134. ) )
1 3 / £ £ / £ 5 7 £ / K3 £
£¤
¤
£¤
7 7,5 7X 7 ( 7X! " 7 7<5 7 ¢ $ , 3 , 135. ) )
1 / ¡ 3 ¢¢ 7 3 £ / / ¢ ¡ K3 ¢ / £ / ¢ 7<5 3 £ ¢¢ / ¢ £¡ E3 / £ K3 £ ¤ 7 7( 7X 7 ( 7X! " 7 7 ( 7 ¢ $ 136. , 3 , ) )
1 ¢ ¢ 3 / ¡ 7 £ / / ¢ ¡ ' ¢ / £ / ¢ £ ¢¢ / ¢ £¡ 7<5 / £ K3 £ ¤ 7X 5 7 7 ( 7X! " 7X 5 7 ¢ $ , 3 ) , 137. )
1 / ¢ 3 3 £ £ / ¢ ¢ / ¢ E3 £ 5 7 ¢ ¡ X7 ( 7 7 ( 7X! " 7X ( 7 ¢ $ , 3 ) , 138. )
1 3 / ¢ £ £ / ¢ ¢ / ¢ £ 5 7 ¢ ¡ 5 5;
/ ¢ ¢ ¤
/ ¢ ¢ 3 £ £¤
139.
140.
( 7X! " 7 5 $ )
1 £ / 3 3 ( 7X! " $ )
1 ¢
3
3
7 5 7 7 , ¢ 7 3 7 £ / ¢ / ££ 7 / £ / 5 7 ¢ £ £ ¢ / ¢ K3
/ K3 7 ( 7 ¢ 7 ( 7 7 , 3 ) , 7 / 4£ 7 £ / ¢ / ££ 7 £ / / 5 7 ¢ £ £ ¢ / ¢ '
/ K3
7
:1 ,
¢
, 3 )
7 5 7 ( 7X! " 7 5 7 ¢ $ , 3 ) 141. )
1 / ¢ 3 3 £ £ / ¢ / ¢ 3 £ 5 7,5 3 £
£¤
£¤
7
,
7
¢ /
£ /
¤
7 ( 7 7 ( 7X! " 7 ( 7 ¢ $ , 3 , 142. ) )
1 3 / ¢ £ £ / ¢ / ¢ ' £ 5 7 / ¢ £ 7<5 £ / K3 £ £ ¤ 5 £) ¢ 5 £! ¢ 7 ( 7! " $ 5 7 143. , 3 , ) ) 7 £ / ¤ ( 7X! " 5 £) ¢ 5_7 ) , 144. $ 5 7 3 ) £) ¢ , 7 ¢ / £ £ / K3 7 £ / £ ¤ 5 5:
145.
146.
( X7 ! " O7 7<5 7X 5_7 )
$ 7 7,5 7 £) 3 ) 7 ¢ 3 / £
£! ¢ , ¢ 7 7 £ , 7 / ¢ E3 £ /
( 7X! " 7 5 7 _ 5 7 )
$ 7 5 3 ) 7 £ / ¢ 3
¢¢ / ¢ £¡ ¢ ¡
/ 7<5 / £
£! ¢ , 7 !£ ¢ 7 / £ £ ' 5 7 ¢
/
7
,
£ / £ K 3
7
/
¤
£ ¢ / ¢
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6.2.2. Series containing ψ(ka + b) and trigonometric functions
" 1. $9% . X+ 1 1 \
2.
& & \
N " $&% ! 3 8
7 G F 7#7 ( \ !
&
& ¢ & £ \ £
3
3
8
\ C £ £ ; 7 #7 ( N !
&
\ £ \
£
B
B
6.2.3. Series containing products of ψ(ka + b) 1.
2.
3.
¢ 3 ¢ 3 = 1 " ¢ 3 C ¢ 3 X 3 C $ E 3 <3 7 1 3 & ¢ 3 £ & ¢ 3 3 £ = 7 7 ( 7 ! " 7 3 C , £ 3 C £ 3 & £ 3 ¤ $ E ) 1 5 5<
E 7 (
C
£
C
7A@CB
7A@CB
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
7 7 7 C £ 3 C $ E 1 ( 7 ! " ¢ ¢ , 3 ¡ C ¡ $ E 1
7
1
$ $ 5_7X!
£ C 3
C
:1 , 3
= = @ O!C@CB £ £ £
3 7 1 D 7 3 C £
= = @ B 7 !C@MB ¢
C $ E 3 7 ) C , 3 # ¤ 1 7 $ E C C <3 <3
7 H ¤
1
7
C
] ¤
$ 3 7 C 3 C , ) 1 ( 7! " 7 G 7 C £ £ , 3 & C £ £ $ E IF 1 7 7 £ £ £ £ G C 3 C $ E IF 1
7
7 1L" J $ 5_7X! E F G , £ CO ¤ 9 $ %
] <3 ¤ £ &3 = = @ C7 OC! @CB
= = @ O !C@CB
7 1 L " $ 5_J 7X!A $ 5 ! F G 7 C £ C 3 ¤ 9 $ % 7 1L" J $ 5_7X! % F G 9 $ % £ 9 £ £ 3 ¢ C C £ 9 £ £ 3 ¢ ¤ m !" 3 0f & $ % E fI = m @ B m 5 5L
m !" $ & % fI O
16.
1 L " $J _ 5 7X! % J( 1L" 5_ 7X! % $
17.
18.
7 F G
7 F 3 G
3Pf O 43fI <3 fI 3 h 3 fI = m ! m V( 7 V ( &B B B@B
¢ , 3 ¢ 7 3
7 1L" $J h 5 ! % F G £ C £ C £ 3
19.
7 1 L" $ 5_7XJ ! % $ 5_7X! F G
C
C
3
£ ) C 3 3 £
20.
7 1L" J $ 5_7X! % E IF G 9 $ %
21.
C
7 1L" &L" $ J 5_ X7 ! % J $ 5n ! % F G 83 # C £ 3 9 £
22.
£
£ C £ ¢ 3 £ ¤
C
£
C
,
£
3
3
¢ 3 C
£
£ 3
¤
£ £ £ ¤
3 , £ 3 ¢
£ I P3 0 ¤
£ 3 ¢ , £ 9 £ £ 3 ¢ 4 R ¤
3
F G 3
£ £ £ 3 ¢ C £ ¢ £ £ ¢ ¤
3 3
6.2.4. Series containing ψ (ka + b) 1.
2.
7 7 ( 3 ¢ 3 ¢ 3 1 ¤ ¢ 4 3 3 1
5 5P
=
7A@CB
:1 , 5
£ $ 3. <3 ¢ 3 1 ¢ £ ¢ 3 3 3 < 3 & 7 £ $ 4. == 1 £¤ ( 7! " 7 $ <3
5. 1 7 $ 5_7X! ¢ $ == 6. 1 $&% ! ( n 5
! X 7
% 3 3 7. $ 1 I5
5
3 7 7 ¤ G 8. $ 5_7X! F J 1 L" 7 /
$&% F G 7 9. 3 ( 7 ( 1L " 7 J $ 5_7X! F G / 3 & / 9 $ % 10.
11.
12.
13.
"
7 1L" $ _ 5 7XJ ! % $ 5_7X! F G 5 / 3 / ¢ 3 m !" 0f $ & % fI m m !" 3 & $ % I f fI m 5;9N
5
/
/ ¢ 3
3
f £
C
= m
7A@CB
@ B 7X O!C@MB
@ B 7 !C@MB
=
=
=
7A@CB
7R@CB
@CB
3 ¢ & /
= m
=
7 m !
=
7R@CB
V( 7 ( &B BB@ B
43fI 7 m V ( 7 ( &B B B0@ B !
!>
!>
6.3. The Hurwitz Zeta Function ζ(s, z) 6.3.1. Series containing ζ(k, z) 1.
2.
" ] ! ' (! 5:' 1 &
$ $ L 3 ; J
)" " & -21 3 ' ( $ 3 1 & <3 '65_7 ¢ 3 0& C = $ (*7X! % 1 ( ! "' O K( 4! 58'! % & ' % -; $
7 ' O ! 3 ' % J $ L&3 7 & -21 ' ( 4! )" " 3 ' % KJ $L $ 5_7 a 1 0& ¢ <3 ¢ 0 ¢ 3 & C = '65_7X! % <
-; @ @CB = !
= @ @CB !
6.4. The Sine Si (z) and Cosine ci (z) Integrals 6.4.1. Series containing Si (ϕ(k)x) 1.
2.
3.
4.
5.
( 7X! " 5_7X! 4 £! $ $ 1 ( 7! " m 5 ! $ $ 1 ( 7! " $ $ m ( ! 1 ( 7X! " $ (:7 $
$ 5_7X! " 0 $ 5 7! m 5 ! D
=(
¢ <3 £
3
3 7
m
3
£! ¢
F m G 3 5;
m
F m G
F m m G
F m G
m
CH
X @CB
=(
@CB
=(
@CB
=(
@CB
=
@CB
6.
7.
8.
9.
7 $ 5_7X!A $ 5_7! $ 1 ( 7X! " 5 m F $ 1 ( 7! " $ $ 5 m F 1
4
£) ¢
:1 5) ,
3
f G 3 7 f
m
3
f G
7 7X m 3
f9
0 f 3
( 7X! " $ 5_7X! $ 5 7! 5 m
3 £
C
=
=(
£) ¢ f
@CB
@CB
f 0 = ( f m =(
@ B
X @CB
6.4.2. Series containing ci (ϕ(k)x) 1.
2.
3.
7 ( 7X! " f < 3
9f
H D 5
$ m 1 7 0 f f f f C 3 m ( 7 ! " £) ¢ $ 5_X7 ! 4 3
0f f 0 = ?m £ ¢ 0 C <3 7 )
( 7! " £! ¢ 3 ( : 7 A ! 5 ! $ $
" $ 5_7X ! ¢ D 4. 4 3 _ 5 X 7 ! 5 $ m 3 C 0f 3 f m $ 5_7! " m 5 4 £) 5. _ 5 X 7 ! $ ] C 3 £ £ f
£! ¢ 0
=
X @CB
£) ¢ 3 7 D C £ H ¤ 7 f <3 £! ¢ f H (:7X! @CB m 5_7X! C = hm
¢ 3 !£ ¢ 0 £ f D 3 F m G m m 5; ,
@CB
=
F m G9H
@CB
. 9 0 9
$ 5_7! " m ( 4 £) ¢ 3 6. 5 _ X 7 ! $ 3 £ f F G ] C 3 £ m
£) ¢ 0 £ f m D m 3
F m GIH
=
@CB
6.4.3. Series containing Si (kx) and trigonometric functions
$
!
( " S -
(87! % (:7! $ S - -21 ¢ £ ! 43 % 3 = >?7 ! ! ( " ¢ S ] (:7X! % (87! - -21 ¢ ¢ £ a ! 43 % 3 = >?7 ( (
¢ 4 3 S # 1. 1 ( " "" Y 4 3 ¢ S 2- 1 ] ( $ ( ! % 43 ¢
2.
$ ( "
( $" ! ¢ 3 4 3 $ 1 S S - ( 7X! " "'& 3 $ 5 7! % $ 5 7! Y £ 3 £ 3 £) 3 £
!
!
"'& 5 7! % $ 5 7! £ 3 !£ 3 £ C@ B (
SEa &
(
@CB
6.4.4. Series containing products of Si (kx) 1.
2.
( " ( 7( ! " ¢ S 3 # (:7! % $ 1 S - ( 7X! " "'& S - -21 ¢ 3 $ 5_7X! % $ 5_7X! 43 Y ¢ 3 £ a - SEa & £ 3 £ 3 = >?7 ! ( ( 7' ! " 3 7 $ 1 1 1
(:7X!
& 5 7! % _ 5 7! £) 3 £ ( !
!
'
(
@CB
B
6.5. The Fresnel Integrals S(x) and C(x) 6.5.1. Series containing S(ϕ(k)x), C(ϕ(k)x) and algebraic functions 1.
( 7X! " 7 " £ $ $ : E ( 7X!
5;93
<3 E
=(
@CB
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
:1 ;
( 7! " $ $ m 5 E 3 F G &4
1 m ( ! m * m , 3 * m , = ) ) " ( 7! $ $ m ( E F G &4 3 m m gF m G 1 =( $ 5_7X! " £) ¢ $ 5 7! E4 3 * $ 1 = " $ 5n7X! ¢ $ (*7X!A $ 5 ! E4 £) ¢ 3 £ = ( 4 3 ( 7! " 7 ) £ ¢ (
E 4 * = _ 5 X 7 ! $ ( 7! " E4 £) ¢ £ 3 _ 5 X 7 9 ! _ 5 X 7 ! $ $ $ * 1 =( ( 7X! " 7
G =( m E f $ 5 m ! gF f
1 ( 7X! " $ 5 m ! TF f G $ 1 7 3 £
f9
f 3 m f = ( m ( X7 ! " $ 5 7!A 0 $ 5 7 ! 5 m !
( 7X! " $ $ m 5 1 3 7 " m
*
£) ¢ f
m
5;!5
) *
@ B
@CB
@CB
X @CB X @CB
X@CB X @CB
@CB
E f m (= X @CB
m , ) * ( m , =
@CB
. 9 0 9
12.
13.
14.
15.
16.
17.
18.
19.
20.
( 7X! " 7 " 3 $ m ( m $ 1 ( 7X! " $ $ (:7
$ 5 7! " 5_7! $ $ 1
"
£ 3 7
m F m G =( =(
4 £) ¢ £ <3 3 *
@CB
@CB
$ 5_7X! ¢ $ (:7X!A $ 5 ! 4 £! ¢ 3 £ 4 3 ( 7X! " £) ¢ 7 "
4 =( $ 5 7! $ 5_7X! ¢ ! £ ¢ 5 7 ! =( 4 3 * 43 $ $ 1 ( 7! " 7 G =( 3 m f F f
$ 5 m ! 1 ( 7! " $ 5 m ! F f G $ 1 7 £ ,
f9 3 f 3 m f m * ( 7X! " 5 m ! £! ¢ f 5 7 A ! 0 5 7 ! $ $ =(
= =
X@CB
X @CB
=(
@CB
@CB
X @CB
@CB
m f X@CB
6.5.2. Series containing S(ϕ(k)x), C(ϕ(k)x) and trigonometric functions 1.
2.
( 7 ! " $ 1 ( 7 ! " $ 1
3
*
3 7 * 5; ;
=
=
5
@CB
5
@CB
! !
( " $ ! ¢ S 3. 3 $ 1 * ( " "" Y 43 ¢ S -21 ] ( $ ( ! %
:1 ; ,
( " (:7X! % 8 ( 7! " S -
5 7! % $ 5 ! $ S - -21 ¢ ¢ 43 43 ! % £ 3 £ 3 !£ 3 = >?7 ( !
!
£
@CB
( " ( " $ ! ¢ ¢ S $ E 43 (:7! % (:7X! 4. S - -21 ¢ ¢ £ ( 7! " " S -
$ 5_7X! % $ 5 ! 3 ! % 3 a - SEa & Y £ 3 £ 3 £! 3 £ = >?7 ! ( ! ( ( @CB ( 7 ! " 3 " 5 @CB = 5. $ ! 1 ( 7! " 3-" 5 @CB 6. = $ ! 1 ( " ( " $ ! ¢ S 43 (* O! % ( ! 7. $ 1 " S -
! 5_7X! * $ % $ ( " "" S Y 3 ¢ S ] (* $ ( ! % 3 ¢ - -21 3 ¢ ! % £ 3 £ 3 £) 3 £ = > ! ! ( @CB ( " ( " $ ! ¢ ¢ S 3 ( O! % ( ! 8. 43 $ 1 S - ( 7! " " S - -21 ¢ ¢ £ a SEa 3 * $ ! % $ 5 7! 43 ! % 3 - & Y £ 3 £ 3 £! 3 £ = > ! ( ! ( ( @CB
4 3 1 3 *
5;9:
. 9 0 9
9.
10.
11.
12.
$
( 7! " $ (:7X! E 7 7 3 3
"
=( 7 3 "
( 7! " $ $ 5 m ! 1 m m * m , )
X @CB
- ) * m , = ( @CB F G 4&
( 7! " $ $ 5 m ! E 3 1 m
3 ! m A ) * m , 3 $
- A ) * m , =(
( 7X! " $ (87! 7 7 3 E 3
( 7X! "
8 E 3 13. ( 7 ! $ $
3 7 £ 7 £
3
$ 5_7X! ¢ 5 7! 14. 4 3 $ $ 1 Y £! ¢ E4 £! ¢ 4 £) £ D <3 E <3 $ 5 7! " 5_7! 15. $ $ 1 Y £! ¢ E4 £! ¢ <3 £) 4 £ 3 ; E <3 83 5; <
" =(
<3
@CB
X @CB
F G &4
=(
@CB
¢ 4 £! ¢ 7 ( @CB H = ¢ 4 !£ ¢ £ ; / £ = X@CB
:1 ; 3
$ 5 7! " 5_7! 16. $ $ 1 Y £! ¢ E4 £! ¢ 4 £) ¢ 4 £! ¢ £ 83 £ ; £ £ 3 ; 3 * = ( 7X! " 17. $ 5 7! Y 4 £) ¢ £) ¢ £! ¢ £) ¢ " =( $ 5_7X! ¢ (:7X!A $ 5 ! 18. 43 $ Y £! ¢ E4 £! ¢ 4 £) ¢ 4 £! ¢ 3 £ E £ £ £ = ( " $ 5_7X! ¢ (:7X!A $ 5 ! 19. 43 $ Y £! ¢ E4 £! ¢ <3 4 £) ¢ 4 £! ¢ 3 £ E £ <3 £ £ = ( " $ 5_7X! ¢ 5 7! 20. 43 $ $ 1Y £! ¢ £! ¢ £) ¢ £! ¢ £ E4 3 £ < 3 3 4 4 = ( 21.
( $ 5 Y
7X! " 7! £! ¢ E4 £! ¢ <3
X@CB
X@CB
@CB
@CB
X@CB
4 £) ¢ 4 £! ¢ 3 7 = ( @CB *
6.5.3. Series containing S(kx), C(kx) and Si (kx) 1.
2.
( 7 ! " $ 1 ( 7 ! " $ 1
3
3 "
*
5;9L
=
=
5 !
!
5
@CB @CB
6.5.4. Series containing products of S(kx) and C(kx)
1. 3
2.
3.
4.
5.
( 7 ! " $ ( & 1S - ( 7! " "'& S 5_7X! % $ 5 ! $ Y £ 3 £ 3 £! 3
( 7( ! " $ 1 S - ( 7X! " "'& 3 $ 5_7X! % $ 5 ! Y £ 3 £ 3 £! 3 ( ( 7"! " $ 1 S - ( 7X! " "'& 3 $ ! % $ 5_7X! Y £ 3 £ 3 £! ( ' 7 ! " $ 1 ( ' 7 ! " $ 1
E 3 1
4 3 ¢ S - -21 3 £ =
( " (87! % (:7! & ¢ 5_7X! % 5 ! ¢ 3 £ a - SEa & >h7 ( ( ( C@ B !
!
( " ¢ S 3 (:7X! % (:7! & S - -21 ¢ ! % 5 7! ¢ 3 £ a - SEa & 3 £ = >h7 ( ( (
3 ¢ S S - -21 43 3 £ =
!
!
!
( " ( O! % ( ! & ¢ ! % 5_7X! ¢ 3 £ a - E S a & >_ ( ( (
"' F G 1
4&
7 1
3 F G 1 1
'
!
!
1.
2.
" ! $&% " - & # -
B
B
'
k, z)
&
¢ ¢ & ¤
7 ! " 1 & )3 ¢ -21 - 3 )3 ¢ X T3h63 ¢ )3 )¢ & ¤ 5;9P
@CB
6.6. The Incomplete Gamma Function γ(ν, z) 6.6.1. Series containing γ(ν
@CB
!
3.
4.
5.
6.
7.
8.
:1 :1 ,
( ]! " ¢) $9% ! " & ¢ £ Q3 ¢ £ & <3 £ & H £ & £ 1 1 C7 ( ! " 1 & - 63 & $ % )3 " W . Z
! " & ; £ - & $ % V- +
-21 m ( ( O " " 1 ! ! £ 1 ¤ m ! " & $ % & N 8 ( 7 5 7
m
" . & & & ¤ 5_7X! " $ 5n !Ij + - - l ¡ ( 7! " ¢) & ¢ 3 - F PG & $ % Q 3 F £ - P3
& H £ & ¤ =
@CB
¤
5_7 G
- \¢) Q ¤
6.6.2. Series containing products of γ(ν + k, z) 1.
2.
7 ! $&% " ] - & N
7 ! " - & 1 3 & & $ %
5 1 ! 5_7
( ¤ 5_7
(
¤ N 5 7 5 7
1
!
6.7. The Parabolic Cylinder Function Dν (z) 6.7.1. Series containing Dν
"
1.
$9%
2.
( ! " $&%
nk (z)
. a & ¢ £ - + a 1
and elementary functions
¢ £ - & 3
5: N
4 ] I5_7X!
F I5_7 G = 7 ! = @& B 7!C@MB # C7 (* 4! F 7 (* G 7 @ & B O !C@MB = ! =
6.8. The Bessel Functions Jν (z) and Yν (z) 6.8.1. Series containing Jnk+ν (z)
1.
2.
L a & J 5_ 7X! ¢ !£ a 4 3
"
* ( !
a 1 #$ 1 ! (
#$ -Q1 a 1 a &
a 1 5 1 & ! (
" W $# 7 ( J $ L5 m ! a 0 & m J L 5_7X! N m 5 ! 7 5 7 ¤ & $ % 1
W - a 1 ( #$ m !" m £) 5n7X! F ,G N !
& a 1 5n7 ( m 5n 7 ?fI $9% a &
W ) : ( 7 ( ¤ ( g 5 7 ! " £) a & J L ! N ! " ! ;
1
5 1L " J 5 7! " a & $9% ( (* 5 (* ¢ F G ) , ) , = ()(:7X! " 65 !A 65 5_7X! " £! ¢ 565n ! " a 0& ! 43 a a ( #$ Y 65 N -Q( 1 1 a 5_ 7! 1 & 1 #$
a a a a
1 & 1 ! ( 3 65 5_7 & N
a 1 5 1 a a & &
W
#$ ! " ( 1 L" ! £) J L ! N ¤ a ; $&% 5 J
&
5 a
1 1 1 1 J L"
5:1
3.
4.
5.
6.
7.
8.
W #$ ( ! ¤ 1 N 5_7
& (:7 (* & N ( 3 (:7 & N
-Q1 1
¤
¤
X@CB
¤
:1 L1 ,
1 ( L" 1 5 L" 1L" ¢ J
C7 ( J ! " C7<5 ! J "
9.
a 1 & $9% / £ F - G ! " m ! " ! " £) ¢ ! ( 5_7X! " ( 5_7! " a & 10. 43 ; $ % m J L ! #- N 5 1 ( O 5n( 7m O ( ( 5:7 5:! 7 m
5 O ( m ( 5:7 ( 1 N
( m 5n7O ( ! 5:7
F G ¤
m !£ ; &$ %
11.
!" !" ( m 5_7! "
¤
a &
W " " ( m 5 1 ( #$ ¤ ! ! 1 N 5 * 7 ( _ 5 m 1
W #$ ( a 0& J L ! N & ! a &1 ¤ 1 1
L " ) ; J & $&%
12.
6.8.2. Series containing two Bessel functions Jnk+ν (z) 1.
2.
3.
4.
5.
1
a &
'&
5_7 5 ( & ! J L 5n O! N n 5 n 5 5 ¤
n
&
7 Z ¢ a Q a . & £ 4 ¤ 4 3
1
+ 1 £) ¢ ¤ a & 1
£) 7 £ £ ¢ ¤ a & & 3 &4
¤ a Q a & ] M E( ! -21 Q & <3 Q -21 0 & 5:,
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
a & 0& 0& 3 -21 & a 1 0& 3 - & 2 - 1 O ¢ 7 ¤ 3 &
( 7! " $ (:7X!A $ 5 ! & a 1 0& £ & H1 £ & 3 1 £ & H 7 Z ( 7! " a & 83 £ 0 & a _ 5 X 7 ! $
- -21
1
7X! " &$ %
Z a 0& +
! " L J $9% a & 5_7! 1 N $& % ! " a Q a &
¢ 4 3 ¢ 4 3 7 m !" C7( m ! " & & ¢ C7 ( m ! " ! 0& 7
4 3 m "
17.
0& ¤
£ & ¤
¤
£ £ ¤ 5 1 ( ¤ 5_7 ! 5_7 " ! F 5 G & ¤
( m ( ¤ 1 1 N 7#7! ( m ( 7 & 1 N 7 1 #! 7 ( m ¤ #$ 7 ( m !" O & m ! ( ¤ ( m ! " a 0& N &
1 & & ( m
£) ¢ m ! " ¢ ( m ! -21 H £ & ¤ 4 3 ( m ! " a 0&
1 1
m !" £) fI $&% a & ( 5 1;- ! ( ¤ m 5_7X! F ,G N
& 5 7 5_7 ( m 5_7
7
16.
.
- &
5: 3
18.
:1 L1 ,
! " $ 5 $&% a 0& 5 ! : ( 7 $
7 ( L 8 ¤ ! ( 7J! ! 5_7! 1 N 5_7 5 &
19.
! " ¢ $ 5 $9% a & 4 3 5 ! ( : 7 $
5 1 ( #$ ¤ L J
! : ( 7! ! 5_7X! 1 N 5 7 5 &
¢ C 7 ( ! m ! " 4 3 m " - 0 & a & m ! m 5 ( m 5 ( ( 1L 1! 7 0& £ !- & ¤ J m 5 ! 5_7X! N : ( X 7 ! 5 ] 5 7
m m 1
20.
21.
¢ m &$ ! % " a & 4 3
65 ( m 5_7! & _ 5 7 ! Q5 5_7X! )( m 5 7! a - a 1 a - 5_7 ! ( ¤ Y F G a N 5_ 7 65 5 7 )( m 5_7
&
5 & L J1 ( L" ¢ J 5_ 7X!
43 & 5 L " a & - 2- 1 & J
5 7X! " g5 ! " 65 ( m _ a 0 & a 0& m !" & $ % &( 65 J 7L ! 5_7X! N
& ! ( _ 5 7 ! " " m L J $&% m ! "
a & 5_7! N &
22.
23.
24.
25.
-21 -21 £ & ¤
a a 1m ( a ¤ m 6 5_7 5_7 ! ( 5 1m ( ( ¤ 5 7 ! _ 5 7 m _
5 ( ! ( n 5 7 ! " " m L 1! ¢ ¤ a 0& J ! N m !" $ 9 % 4 3 ; n 5 7
m
1
5:95
26.
¢ )( ! " 9$)% ( ! ( m 5_7! " a & 4 3 m " - & 5 7L m 5 & - L m ! a - _ J J
F 5 7 L m 5 ! m 5 - L ,G 5_7! )( 5_7X! a _ J
J a a m 5 1 & - ! ( Y N
7 g5 5_7 5 m
& 5_ 7 Z ¢ a Q a . & £ 0
4 3
1
+ 1 a 7 7
1L & 3 F G 0& J a
& 1 J L Z
£ K
7 J a 1L 7 ¢ a
0 & 3 F G & 4 3
& 1 J L 7 Z1 £ 7 ( K
27.
¤ ¤
28.
¤
29.
6.8.3. Series containing three Bessel functions Jnk+ν (z)
¤
1.
¢ a 0 & £ 4 3 - 0 & & 7 L & £ & J 5_X7 ! N &
2.
¢ a 0& 4 3
£ - 2- 1 0& a 1 & L ! J 5 7X! N &
5 1 5 1 5 & ( ! 5 7 5_ 7 5 1 _
#$
1
¤
5 1 5 1 5 & ( #$ ¤ !
V 5 7 # 5 1 &
7 Z1 M ! H £ ¤ a
Q 0 & 7 ( a
1 1
3.
5:;
4.
:1 L1 5
7 Z1 M ! £ ¤ a a Q Q & 7 (
1 1 1
6.8.4. Series containing four Bessel functions Jnk+ν (z) 1.
2.
7 7 & N & 0&
N & 0& a 1 &
( 1 1 ¤
7 # ! 7#7 & & ( ¤
!
3.
7 Z1 7 a 0& 7(
Q a 1
1
4.
7 Z1 7 £ H £ ¤ 1 a 1 Q 0& a 1 0& 7(
5.
7 Z ¢
a . & F £ 4 3 Q
a + 1 1
6.
7 Z a Q a 0& 3 F £
& 4&
7.
7 7 Z £ & Q 0&
Q
8.
9.
1
a Q
- Q
a &
H
£ H £ ¤
G
H
G
£
H
£
£
¤
£ 4 ¤
4 ¤
- &
7 3 7 Q 0& 7 Z1 £ £ ¤ 7 (
a a 1 Q
- 2- 1 Q
a a 1 & - -21 0& 7 £ £ ¤ 7 Z1 7(
5: :
10.
11.
¢ 4 3 1
a Q 3 7
0 0
1 1 7 Z F" 1
- Q
- & 7 Z
Q & £ £ ¤
7 & 3 1 0 & 0 3 £ 1 G F " & 3 £ & G ¤
a &
6.8.5. Series containing Jk+ν (z) and ψ(z)
"
1.
2.
3.
L $J _ 5 7! % fI 0& fg3 ¢ 1 0 & 3 fg3 ¢ F G ¢ ¢ £) ¢ a & £ 4 3 3 " &
1 3 " £ & <3 ¢ £) £ a & " £ & 4 3
&4
3 " £ & <3
¤ - 0&
£ &
C
£ &
C
¤
¤
6.8.6. Series containing Jν (ϕ(k, x))
S
& 2 1 S 1 1 ( 7! ( &(' & ( 7
a 1 O '65 7 $ : O'Q5_7! % £ S 2- 1 2 &- E S a 1 J $ LF = S 2. - &-21 &SEa 1 1 '& a O'6 ( 5 7! 7! % £ S -21 &- SEa 1 1 O'6$ 5_7 L F $ 3 J = 1.
5:<
G 3 S F G O' @CB !
G F G S O' @CB !
3.
4.
:1 L1 :
S - a 1
1 ( 7X! '%& '! % £
S- -21 ' $ 5_7 S -21 2 &- S -21 * 3 G S F $ L F J = O')(:7 ! ( 7X! '%& £ SEa S ¢ a O'! %
&-21 , &-21 - 1 &S -21 4 3 1 O' Y ']" )" " $ L $ L F]3 G F G = O')(:7 ! S J J
C@ B
S -21
5.
G
@CB
¢ S - a 1 S 4 3 - ( &('-2& 1 1 ( 7 ! O'! % £ SEa &-21 &-21 - S -21
O' $ 7 ( Y ']" )" " $ L L F G G = O')(:7 ! @CB F S J J ( 7! ( &(' & £ SEa a S ¢ O'65 7! %
1 , a 1 - S -21
- &-21 1 - S 43 1 a '])" "'& O'65_7 $ Y 1 " , G G $ L L F 3 F = ' ! @CB S J J ( 7! ' & £ SEa a S ¢ - &-21 &SEa 1 '65_7! % 1 a 1 - S -21
43 1 a '65_7 $ 7( @CB Y 1 '# )" " "'& ' L F G G $ L = F ! S J J
6.
7.
8.
9.
S a 1 &S
2 1 1 '%& ( '7X! ! % £ S 2- 1 2 &- S -21 $ ' L F $ 3 J = " $ ¢ J L 5_7X! 3 7 <3 4 3 8 ( 7 $
=(
5: L
G F G S O')(:7 !
a 1
!
@CB ( @CB
10.
1
( 7 ! " $ 5 m
F
" f G 3 m
0f =(
)£ ¢ S £! ¢ - &-21 1 - S ( &(' ( '67! 5_7X! % £ S -!&4 a 1 - S -21
'g5 7 £ £ a $ Y a G G $ L 3
= F 3 F & S &- 1 J
( 7 @ B
!
11.
12.
13.
14.
15.
16.
O'
!
)£ ¢ S £! ¢ - &-21 &SEa 1 7! ' £ S O '6( 5_ 7X! % -!&4 a 1 - S -21
'g5 7 £ £ a 7( $ Y a G G $ L F 3
F &S &- 1 J = O' ! $ 5 7! " $ 5_7X! 4 £) ¢ £ a 1 <3 H $ 1 =( ! " $ 5_7X! ¢ 5 7! 4 £! ¢ £ a 1 <3 43 $ $ 1 =( ! $ 5_7! " £! ¢ $ 5 7! m 4 " &£ £ fI F G 3 L fI m !& & fI m H fI = ( X7 ! " )£ ¢ 5 7 f -
$
!
£) ¢ f
=
5: P
" m
!
@CB
@CB
( X@ B
( X@ B
( @CB
0f ( @ B
( 7X! " 7 ']" DXF G 17. $ 5 1 L J ']" 3 ¢ &-21 & -21 $&% O') ( m m
:1 L1 <
7 G f
f H -
F
, ) * ! ")" $ ( O! % &- - a fI = 7 ( O'I@CB
6.8.7. Series containing Jν (kx) and trigonometric functions 1.
2.
3.
( ! 7& ! " $ 1
( 7 ! " $ 1 ( $ 1
$ !
& "
3 '&
5 7! =
L 5 7! 3 J _
5
5
!
!
( X@ B
=
!
!
( 7 X@ B
" S - S - L J 5_7X! " F G
$&%
- -21 ( ! £ £ £! £ ! % 3 3 3 = >?7 ! ! ( C@ B ( 7! " ( 7X! ( $ ( & " # (:7X! % 5 ! F G SEa -
1 S -
a ( 7! "
5 _ 5 7 ! G 9 $ % $ 3 F S ( X7 ! Y - -21 ! % ¢ 3 £ a - SEa & £ 3 £ 3 £) 3 £ = >?7 ( ( ( ( @CB
( 7! ( L J 5_ 7! # (:7X! % 5_7! ( " ( 7! ( " S Y ] (* $ ( ! % 2 S - !- & 3 ¢
4.
!
!
!
6.8.8. Series containing products of Jν (ϕ(k, x)) 1.
( 7 ! " $ ( && 1
( % 5 S -21 ( 7X! " 5 65_7! 3 $&% $
5<9N
X7 ! ( & SEa g5_7X! 5_7! F G
F G W F G
S ( 7! Y - % 5 5_7X! F G a ¢ 3 =( ! ( " $ L ¢ J 43 $ (87 5_7X!
3 a 1 £ ¢
2.
3.
1
( 7X! " $ 5 m F
4. " $ " 1 m 7 5.
£ a - SEa 1 £ 3 £ 3 £! ( (:7A@ B ( 65 ! ! 7
3
a 1 <3 a ( = ! ( R7 @CB " f G 3 m f
= ( ! ( 7 @ B
( X7 ! " $ 5 m F
f G £ ¢ a 1 f - a 1 f 3hf
( 7 ! " !£ ¢ _ 5 7 f9 R $
a 0 f &3
f9 f = ( ! ( @ B " £ ¢ f m !
0f
= ! ( R7 @CB
6.8.9. Series containing products of Jν (kx) and trigonometric functions &( ( 7X! " 1.
2.
3.
(& !& $ 1 ( 7(& ! " $ 1
L 5 7X! 3 65 J 7! _ = 5
!
&
L 5 7X! 3 6J5_ 7! _ = 5
!
65 !
( 7A@ B
!
65 !
( 7A@ B
&(
!
$ ! ( && " J 7XL ! 5_7! 6 _ 5 $ 1 ( 7! ( ( " 65 5 7! Y 65_7X! ( " 5_7! ( " g5 ( 5_" 7X ! ( " S -
S -
¢ 65 5 7! " ( " "" S
65_7X! " 5_7! " 65 5_7X! " F G 43 2- 1 # ( $ ( ! % $9%
5<
+
4.
:1 L1 N
S - -21 ¢ ¢ ! % £ 3 £ 3 !£ 3 £ 4 3 4 3 = >?7 ! ! &( ( 7X! " L ( && " 65_ J 7X! 5_7! $ 1 ( 7! ( 65 5 7! Y # (:7X! % 65_7X! ( " 5 7! ( " ( " 6 5 5_7! ( " F G S -
( 7! " 65 5_7X! " 3 $9% 65_7X! " 5_7! " 65 5_7X! " F G
S ( ! Y - -21 ! % ¢ 3 £ a - SEa & £ 3 £ 3 £! 3 h > 7 ( ! ( (* ! 65 = !
( @CB
S -
£ !
7 (* @CB
6.8.10. Series containing Jν (kx) and Si (kx) 1.
( 7X! " $ ( & & " ( 1 65 (87! ] (:7! (% 7 ! 5 ! 5 5 ! 5g5 S -
( 7X! " $ 565 5_7X! 3 $9% $ 5 g5 5_7X! $ 565_7X! $ 5 S - -21 ( 7X! a ¢ £ a SEa Y _ 5 X 7 ! 5 7 ! % 2
1 3 -
= >?7 ! (
SEa a -
! F G
a a
5_7X! F G
&
£ 3 £ 3 !£ 3 £ ( ( @CB
!
6.8.11. Series containing Jν (kx), S(kx) and C(kx) 1.
2.
" " 3 5_7 ! = " " (! & X7 ! " 3 5_7X! $ 1 = (!& 7X ! " $ 1
5
5
!
!
( 7A@ B
5< ,
!
!
( X@ B
6.8.12. Series containing Jkµ+ν (ϕ(k, z))
4-4# $ &
1.
2.
$ 1 $ 1
,7 5 7 (8
7 5 m & m
7
W = 1A-
W W ! 1 N 7( m #7< 5 m
7
#$
3 ¢
= g M !
7A@CB
" 3 '" & 8 $ 5 m ! "" L $9% F ,G a / & 3 m J 5_7 !
f - -21 0/ f&
!& L $ 3 "" 8 " $&% a / & J 5n ! ¤ 3. F ,G 1 7 O ]! 7 ¢ 4. 43 & 3 7( 3 7 ( 1 M 5 ! 5. & 7# C7 (8 ! 1 7 6. $ & 1 7 3 7. $ & 1 7 O 7 7O7O 3 8. $ & 1 7 O ) £ ¢ 9. $ 5_7X! a 1 4 & 7 O ¢ ! £ ¢ 10. $ 5 7! a 1 4& ) 3 , 7 7 ¢ ! £ ¢ , 11. $ 5 7! a 1 4& ) 3 7 12. -21 & a 1 & 7(8 3 1
¤
5<93
= g M ]!
7A@CB
= g M ]!
7A@CB
= g M ]!
7A@CB
= g M ]!
7A@CB
= g M ]!
7A@CB
= g O ]!
7A@CB
= g O ]!
7A@CB
= g O ]!
7A@CB
= g M ]!
7A@B
+
:1 L1 3
7 & a & ( m $
-21 1
1 7 3 7 N 7<75 V7 m m & m ( 7 ! "
& 3 14. $ 1 ( 7! " 15. $ ( m & 1 O 7 3 m 0 I f & 0 I f & ( 7! m m m : -R
R
13.
! V7
( m ( m &
= g M ]! = g M ]!
7A@CB
( m &a = h M ! 7R@B
7 # 7 !
N & & &
! " L J 5 7! $&% ; - -21 a 4 4& _
16.
7A@CB
= g O ]!
7A@CB
6.8.13. Various series containing Jν (z) 1.
1 3 7 2.
&
1 7 Z 3 -21 F "
1 7 $ 5 7X! " a 1 Q 4 £) ¢ 4& ] ! " Z ( 1 L 0
J
( 7X! " $ 5_7X! a Q 4 £) ¢ & 1 O ]! " Z 5 1 L
J " ' "
( 7! " $ ' 3 ' 5_7X! 1 1
1
3 £ 1
3.
£
H
32
-21 H £
4.
-!&4
3
G ¤
¤
¤
1
5
!
'
!
( 7 B
6.8.14. Series containing Yk+ν (z) 1.
2.
3.
4.
7 Z 7 a 0& a Q 3 1 7( Q a
1 1
1
7 Z 7 a 0& a & 3 1 7(
Q a 1
1
1
7 a & a 0& a 0& & 1 1
1
7 3
1 & 1 17 Z £ F " 3 1 & G F "
1 &
£ ¤
£ H £ ¤
#& <3 ¢ ¤ & £
3 G
=
@CB
6.9. The Modified Bessel Function Iν (z) 6.9.1. Series containing Ink+ν (z) 1.
2.
3.
0& 1 m !" $&% a $ 5 $ 5 !
& 1 & 0 ¤
0 &
L J 5 7 !
5 1;- ( 5_7 ¤ 5 7 ( m 5_7 ( O
N _ !
! " (87 $9% a & " " " (:7X! ! + -!& . + . - F m ! " ! " ! " ¢ ( 5_7X! " ( 5 7! " a & 43 ; $&% m L J ! - N 5 1 ( 5_( 7m ( ( m
5 ( 5 7 ¤ ! " L ; 1 J _ 5 X 7 !
N 5 7 (65_7 ! ( $&% a &
4.
5.
5< ;
7 3 £G ¤ 5_7 O ¤ 5_! 7
6.
7.
4 3 1
¢ J
J 7 3 F
a 1L & a &L
Z
7 G &
¢ a 0& 4 3
L J 5 7 ! 1 N
.
+
:1 P1 ,
W
!
#$
£
(87 ( 6 _ 5 X 7 ! " L ¢ ! £ J ! ! 4 3 a 0 &
8. " 1N
!" a 1 L " 1 L " !£ ¢ J J
9. 43 $9% 5 1 L" a & L)" a & J J a £ -21 1 J ! & L ( m !" !" L £) J ( 5 7! " a 0&
! - 1 N 1 10. ; $&% m
W !
11.
¤
¤
V+ 2- 1 . F G ¤ m 5 1! ¤ (* m 5_7
" S 2a S $ 5 ! %
3 - S I + -
K
¤
5_7
/ £
a &
( " S S . 43 ¢ $ 9$ I5% ]! ( 3 $ ! % 8 S
; £! & ¤
6.9.2. Series containing Ik+ν (z) and ψ(z) 1.
J( L $9 %
"
fI a 0&
L J 5_7! 0 fI
3 2.
!&
W #$ 7 #7 L J n 5 ! m N & m 5_7 ! n 5 ¤
" J( L $ 5_ 7 ! % fI 0 & 3 fg3 ¢ 1 0& ¢ ¤ fg3 F G - 0& 5<9:
3.
"
J( L $9 %
£) '& L 3 J 5: O!
fI a &
7
m N &
L J 5_7X! fI W #$
7 V7 5_7 ! : 5
7 5 _ m 7 N &
W #$ 7#7 ¤ ! 5 ] a & n
6.9.3. Series containing products of Ink+ν (z) 1.
2.
3.
4.
5.
4 3
43 1
¢
¢ J
J
O ¢ 3 0& 7 3 ¤
a 1L 7 7 V 0& 3 F G V 0& a &L Z
£ K ¤ ¤ m !" 7 7 1 ! C7 ( m ! " & 0& 1 N 7V 7 ( m ( m 7 m !" 7 ¢ C7 ( m ! " V 0& # & 1 N 1 7#7( ! m ¤ 43 ( 5 ¢ J1 mL"J1 mL" J1L" ¢
C7E( m ! " C7 5 m ! " V a & 43 $9% 1 #$ 1 &! N & 7E( m 7 5 m ¤
1 JL ¢ £! ¢ V
O ¤ 4 3 0 & a 1
( V7 #$ ! O " m ¢ ( ! V
N & m ! ¤ 4 3 0 & m " a 1
& ( m & &
( m ! £) ¢ m ! " £ & ¤ ( m ! " # a 0& -21 L 1 - 1
7 $ (:7X!A $ 5 ! 0& a 1 & £ & L1 £ & <3 1 £ & L £ & 0 ¤
6.
7.
8.
9.
5< <
10.
11.
12.
13.
¢ ( ! " ! & a 0& N 4 3 " 1 1 ¢ £) #
7 4 3 & a &4
!& L ¢ J 43 V a 0& 3 5n O! N & 1 ! " $ 5 $&% a & 5 ! : ( 7 $
:1 P1 3
& ( ! ¤
(
£ & 3 £ 3 ¢ ¤
5 7 5 & ¤ _ 5n 5n ! 5n
L : ( 7J ! ! 5_7X! 1 N
14.
¢ 5$ 5 ! (:7 9$ % ! " V 0& 4 3 a $
¢ 4 3
16.
17.
7 (87J! L ! 5_7! N ! 1 5_7 5 5 m ( ! " ( m 5_7X! " L 1 J 5_7X! N $9% m ! " a 0& 5 7 5!
m
& 5 ! " ( m 5_7! " L ! 7 ¤ $&% m ! " # a & J ! 1 N m 1 5_
¤
&
7 ¤
¢ $9% ( 5_! " 7 ! m ! " ! ( " 5_7X! # 0& 4 3 " " a m 5 ( m ( 5_7 L 1 J ! N & 5_7] ( m 5_7] ( ¢ m $&! % " a & 0& 4 3 65 ( m 5_7X! a 5_7X! 6 5 5 7 ! ) ( _ 5 7 ! ,G F m a - a 1 a - 5 Y N _ 5
& 7 g5 5_7 ( m
18.
5 1 #$ ¤ ! 5 7 5 &
15.
!
¤ 5 7
5<9L
7 ¤ 5_! 7
19.
! " " ! F $9% a Q a 0&
5 G & ¤
( 5 & L J1 L" ¢ J 5 7! -21 -21 £ & ¤
43 & 5 L " a & - -21 0& J
£ a 0& - 0& & #$ 5 5 5 7 L 1 & ! ¤ 1 # & £ & J
N 5 7X! & 5 7 # 5 V 5 7 1 1
£ a & - -21 0& a 1 & #$ 5 5 5 L 1 1 & ! ¤ ! J 5_7X! N 7 5 5_ & 1 &
& & ! ¤ a 0&
& N
& 1
20.
21.
22.
23.
24.
7 7 Z £ 0& # Q # & #
Q #
¢ & 4 3
1 1 3 7 7 Z 0 3 -21 F "
1
£
¤
25.
26.
¢ Q & 7 Q & 4 3 7 Z £ 5<9P
1
3 £ 1
£
G ¤
4 ¤
27.
71
:1 P1 5
7 3
1 & 1 & Z F " 3 £ G F " 3 £ & G
1
1 & 1 &
¤
6.9.4. Series containing Ink+µ ((nk + ν)z)
4-4# $ & 7,5 7<5 ( 7 ! " 1. $ & 3 1 ( 7! " a 4 £) ¢ & 2. _ 5 X 7 ! $
1 ( 7X! " ¢ 3. $ 5 m & m
1a
W =
O 7
W W 7 ( #$ 3 1 N ( ! m m
= g M !
7A@ B
= g M ]!
7A@ B
= g M !
7A@ B
= g M ]!
7A@ B
= g M ]!
7A@ B
6.9.5. Series containing products of Ink+ν ((nk + ν)z)
a W = 1 7,5 7<5 M ( ! ¢ 43 & 7 # M 5_7X! 1 ( 7! " $ V & 3
4-4# $ &
1.
2.
3.
4.
5.
6.
1 ( 7 ! " $ V & 1 ( 7 ! " $ V & 1 ( 7! " $ 5_7X! V a 1
( 7! " $ 5_7X! V a 1
3
3
= g M ]!
7A@ B
O 7 3 7 7 3 3
= g M ]!
7A@ B
4 £) ¢ &
= g M ]!
7A@ B
¢ ) £ ¢ 4 & ) ,
= g M ]!
7A@ B
5L N
7.
8.
9.
H
L
7 ¢ ( 7 ! " £ ) ¢ g M ]! A7 @ B V
4 & , = _ 5 7 X ! $ a ) 1
7 7 ¢ & a & 5_7 F ¢ G 3 4 3 # -21
1
1 = g M ]! 7RC@ B 7V7 m 7 7 ( 7X! " ! $ ( m -21 & a 1 & m 3 m N & 7<5 m V7 ( m & 1 = g M ] ! 7RC@ B
6.10. The Struve Functions Hν (z) and Lν (z) 6.10.1. Series containing Hk+ν (z) and Lk+ν (z)
"
J L ¤ 9 $ %
H & & " ( J L ¤ 9 $ %
& L 0&
1.
2.
3.
4.
5.
"
J L H a 0& & L " L a & J " JL $9% H a 1 & " ( J L $9 % L a 1 0& £
"
6.
!&
#$ 1 7 ! ¤ 5 & L N a & a & & J
£ 0 & 3 0& * & L 0& 3 & L & 0 ¤ 1 1 1
& 3 0& & H 0 & 3 1 1
L $ J 5_ 7 ! % H a 1 & 3 *
*
0& H & 0 ¤ 1
& <3 1 & L & <3 0 & L1 & ¤ 5L1
"
7.
+
J( L $ 5_ 7! % L a 1 & *
3 *
:1 N1 ,
& 0& H & <3 1
& H & ¤ 1
6.10.2. Series containing Hν (ϕ(k)x) 1.
" " $ ¢ 4 3 $ m 5 1
m H '&
'&
L
L 3 J 5 =( &L J
!& L J 5 & J L F m G H F m G =(
2.
F m G
F m G
" " $ ¢ 4 3 $ m ( H 1 m 3 '& ! " " ! !& $ ¢ 4 3 H 3 F G H 8 ( 7 $ 5 &
J L =( ! " $ 5 7! ¢ : ( X 7 A ! 5 ! H 4 £) ¢ 3 !& H £ $ $ 4 3 =( ! ( 7! " " " a ) £ ¢ ' & 5 & L 1 $ 5_7X! H 4 J ( = ! " $ 5_7X! ¢ 5 7! H 4 £! ¢ ¢ H 3 £ H 43 $ $ 1 =( ! " " ( 7X! " m '& G 3 H f $ 5 m 3 8 H F f
1 =( !
( @CB
!
3.
4.
5.
( X@ B
H
( X@ B
-21 ( @ B
6.
7.
5L,
-21
( X@ B
( @ B
( X@ B
8.
H
L
( 7X! " $ $ 5 m 3 !& 8 H F f G "m " 1 7X £ ¢ 3
f9 R H 0f 3 f =(
-21 f !
H
( @ B
6.10.3. Series containing Hν (kx) and trigonometric functions 1.
2.
( ! 7& ! " $ 1 1
( X7 ! " $ ( &
3
H
H
S - ( 7! " 3 $ 5 &L ¢J £ Y 3 a =(
3.
( !7& ! " $ 1
H
!
!&
a 1 5
!
( &( "
!
( X@ B
5 5 1L
( 7X! 5n7X! % 2 a 1 3 ( £! 3 £ ( !
(:7 X@ B
&L ( "
!
5
!
( X@ B
( "
" S L L -
5 & L ( " & L " J 5 & L J
J J
S - -21 ( ! £ £ £! ! % 3 3 3 (
J
J
£ 3 ¢ =
( &( $ " ! ¢ H 4 3 $ 1 ( & ( 7X! ( L Y J 5 1 L 5 J
J
( 7! ( L 5 1LJ "'& 'J &
J S - -21 J L$ 5 5 & J SEa L £ £
- & ( 3
L 3 J 5 & J L =
5 &L =
( & $ " ! 4. H $ '& 1 ¢ S L J
5 & L 3 J ( " "" Y 3 ¢ S -21 ( $ ( ! %
5.
J
F G
'&
!
!
L J 5 & J L " S - ( $ 5 5 1 L 3 / J
5L 3
"'&(!& L J
&L $ 5 5 &L
J
7X! "
"
£ (* @CB
+
Y
:1 N1 5
S - 2- 1 ( ! ¢ £ ! % 3 a - SEa & £ 3 £ 3 )£ 3 £ = ( ! ( ( ! (* C@ B
6.10.4. Series containing Hν (kx) and Si (kx)
( ( 7X& ! " H 1. $ ( &( " 1 ( 7! ( L 5 1L J 5 5 1L
&J
S - -21 ¢ J ¢ Y 5 7! % 5 7! 3 43 =
S -
3 £ a >?7 (
!
"'&(!& ( X7 ! " L $ 5 &LJ $ 5 5 &L J J
S Ea & £ 3 £ 3 )£ 3 £ ( ( C@ B
!
6.10.5. Series containing Hν (kx), S(kx) and C(kx) 1.
2.
(!& 7X ! " $ 1
( 7! " $ ( & '& 1 ( 7! ( S - -21 J ¢ Y 3
" " !& 3 H 5 &L J =
5
!
!
4.
( @ B
E H "'&(!& ( &( " " S - ( 7! L L 5 1 L J 5 5 1 L 3 * $ 5 & L J $ 5 5 & L J J
J&
a SEa £ £ £) £ ¢ £ 5_7X! % 5 ! 3 - & 3 3 3 = >?7 ! ( ! ( ( @CB " " '& (!& 7X ! " H 3 5 &L $ 1 J 5 ( X@ B =
3.
!
( 7X! " ( & " H $ ( & " 1 ( ( 7!
S - ( L J 5 1 5 5 1 3 * $ 5 L L J J
J
5L95
!
"'& '& L J
&L $ 5 5 &L
J
7X! "
Y
H
L
& S - 2- 1 ¢ ! % _ 5 7! ¢ 3 £ a - ES a & £ 3 £ 3 )£ 3 £ 3 = >?7 ! ( ! ( (
@CB
6.10.6. Series containing Hν (ϕ(k)x) and Jµ (kx)
( 7X! "
1.
2.
(& !& $ 1
H
'&
L L 3 J 6 5_7XJ ! 5 & L J
= 5
!
!
65 !
4.
!
!
6.10.7. Series containing product of Hν (kx) 1.
( X@ B
7 $ 5 X7 ! Q a 1 Q H £! ¢ & ! " Z £ ¤ 5 1 L 3 2
-21 1
J
7 $ 5_7X! a Q H 4 £! ¢ & 1 O ]! " Z ¤ £ 5 1 L 0 32
-21 H
J
( 7X! " H $ ( && " ( & " "'& '& 1 ( ( 7! ( 7X! " L S L L
J 5_7J !
$ 5 &LJ $ 5 5 &L 5 1 L 6 5 5 1 L 3 J J J &J
S - -21 ¢ L ¢ £ a SEa £ £ 3 £) 3 £ Y J 3 % 565_7X! 3 - & 3 = >?7 ( ( ( @CB
3.
( 7! " $ ( && H H 1 ( 7X! ( " ( &" !& S - ( L L J J $ 5 5 1 L 5 & L 5 g5 1 L 3 J J
J J 5L;
"'&& L J
& L $ 5 g5 & L
J
7X! "
+
S Y - 2- 1 43 ¢ Y £ 3 £ 3 £! J 3 ( ''7& ! " H $ 1 1
L 5 J & L 5 5 £ = J h > 7 ( ! " ' " 3 ' ' 5 J
2.
&(!&
:1
¢ 3 £ a - E S a & &L (
!
a 1 &L ' 1
!
(
!
@CB
( 7 B
6.11. The Legendre Polynomials Pn (z) 6.11.1. Series containing Pnk+m (z) 1.
2.
3.
1
$ _ 5 7 5 7X!;j $ $ _
7 (
=( 7
3 ¢
7 = @ @ B !
$ _ 5 7 7<5 ¢ $ $ _ 3 F G = ( 7 7 ! = @ @ B 5 7X! j 1 $ 5_7 7 ( 7 ( £ £ $ $ 5_7X! j 3 F G
7 ( < 7 5 1 3 F G 3 £ & F G = ( 7 7 ! = @ C7 O!C@CB
6.11.2. Series containing Pnk+m (z) and Bessel functions 1.
2.
3.
7 (* £) ¢ a 1 Q j & ) * , ¤ 43 ¢ ¢ a 1 Q j & * Q& ¤ 43 ¢ a &4 Q j a 1 & * Q& ¤
4.
¢ J 1$9L % " a Q j 0& *
1
5.
&L "
J 9$ %
F
F
a &4 Q j a 1 0& * 5L :
¢ 3h G ¤
¢ 3h G ¤
6.
& '
)£ ¢ a Q j & 7 (8 1
/ £ 3 £ ¤
¢ ¢ a Q a Q j & 4 3
- 1
7 5_7! & ( L " J
¢ J 1$9L % " a Q a Q j & 8.
- 1
7 5_7! & ( L " J
¢ ¢
4 3 9. a 1 Q j 0& 7 1 Q& ¢
10. 4 3 a &4 Q j a 1 0& 7 &4 Q& 2 ¤ £! ¢ j 11. a 1 Q & * 7.
12.
¢ a Q j &
*
1
13.
a Q j a &
1 *
&4
14.
¢ ¢ J 1$9L % " a Q j 4 3 1 5L<
N &
N &
G
£ ¤ 1 Q& 3 Q Q& £ Q
&4 Q& <3
¤ Q&
Q& ¤
0 & F
$# & ! ( ¤ 7 ( V7<5 &
Q& ¤
$# & ! ( ¤ ( V7<5 1 &
1 - 1 N 1
& !
( ¤ 7
+
15.
¢ 4 3
:1 3
&L " J $9% a 4& Q j a 1 0 & F 7 G 4& #- N ! 7 ( ¤ 1 1
6.11.3. Series containing products of Pnk+m (z) 1.
2.
3.
4.
5.
$ 5_7 5_7X! j $ $ 1 $ 5_7 $ 5_7X! $ 1
j C7( j j ¢ 3
7<5
C7( !A C7 ( !
<7 5 3 F
£) ¢ a 1 Q j 0& F
2.
G
=( 7
7<5
F
7A@ B
G
7R@CB
¢ 3h G ¤
£) ¢ a Q - a 1 Q 2 j & 0
C 7 ( ! ! N & £) ¢ F £ 3 ¢ G a Q j 0&
1
3 1F£ 3 ¢ G L F £ 3 ¢G £ 3 ¢G F
6.11.4. Series containing Pnk+m (ϕ(k, z)) 1.
=( 7
! C7,5 ! 3 ¢
1 &! ( ¤
7 5_7 (
L
£ 1F
3 ¢G ¤
$' ¢ $ G j F "
$ 5n 1 ( ' * '65 ' $ L J 1 $&% L " 4 3& F L 3 7 G 3 J ' % 1 L ' & -21
J = R7 @CB $ 5n 5n $ 5_7! " " ¢ 9 $ % $ j $ 5_7X!A $ 5n 5_7X! , ) - - A D F G F GIH ( & 7 B ! 1 5L L
+ ) " * "
6.12. The Chebyshev Polynomials Tk (z) and Uk (z) 6.12.1. Series containing Tnk+m (ϕ(k, z)) 1.
2.
3.
" ¢ 9 $ % & 3h G F 7 ' & a
1 ) $ 5_X7 ! $ 5_7X! ( m " " ']" m
J L ¢ & $ %
D 4 3 " ( 7X! a F $ m $ m ( '%&
1 $ m 1 3 O' ! % m
$ _ 5 7 $ 5 7! ( m , . 7<5 m
. 7 ( m + F G 3 + F GIH ¤ ( G
D m
m H
=(
6.12.2. Series containing Tnk+m (z) and Bessel functions 1.
2.
3.
4.
5.
6.
7.
8.
7 43 ¢ Q & Q& Q ¤ 7 43 ¢ a 1 Q a 1 & Q& ¤ ¢ £) ¢ a 1 Q a 1 & F 3h G ¤ 7 43 ¢ Q & £ Q& Q& 0 ¤ 7 43 ¢ a 1 Q a 1 0& H £ Q& ¤ £) ¢ a 1 Q a 1 & Q 43 ¢ Q a 1 Q 7 Q &
H
£ 2- 1 F
7 a 0 & £ Q& ¤
1 1 ¤ Q 5L P
¢ 3h G ¤
@CB
+
:1 !, 3
7 Q & Q& Q 0 ¤
7 ¤ 10. Q& a 1 Q a 1 & 7 L £ Q& ¤
a 11. Q &
1 a 1
7 & £ Q& Q
12. Q
7 £ Q& Q ¤ 13. Q & 9.
L
£ Q& ¤
6.12.3. Series containing Unk+m (ϕ(k, z))
" ¢ 0 & G 7(8 F 9 $ % h 3 1. D F
( 7X! " a F $ m G
% ' & 2.
1 $ m ( $ m ( 1 3 O'65 7! % a 1 D m m ( 7! " a F $ m G
' &
3. (
1 $ m ( $ $ m 3 1 O'65 7! % a 1 D 7 H 3 ¢ 4f m m 7 $ 5_7
' & a 4.
1 ) $ 5_7! ( $ 5_7X! $ 5_7X! ( m " '#" a L " m . 7<5
1 J 3 ¢ $&% D 43 + F
¢ 3h GIH ¤
=(
m H
&
=(
2.
¢ 7 (* a Q 0 & * 7 a Q 0 & 7 (8 F
1 5P N
m
@CB
. 7 ( m G 3 + F GIH ¤
7(8 ¤ , ) *
¢ 3h G ¤
@]B
m ,
6.12.4. Series containing Unk+m (z) and Bessel functions 1.
3.
4.
5.
6.
+ ) " * "
43 ¢ !£ ¢ 43 ¢ ¢ a
a 1 Q &
Q& ¤
¤ Q a 1 & Q& ¢ ¢ a Q a 1 & 7 (8 F 3 G ¤ h
¢ a Q - a Q & W ! . #$ + ,-Q1 : ! ( V7< 5 ¤ ( 7X!A ( O ! 1 N
¢ Q a 7 Q & £ £ £ ¢ £ £ ¤ C7 (8 ! / / 3 K3 / 3h & / 3 7 £ ¢ 3h G ¤ 0&
8. H F Q 7 8 ( a
1
¢ £! ¢
H £ Q& ¤ 9. 4 3 a 1 Q & -21 ! £) ¢ ¢ (87 !A (* ! 10. 4 3 a Q - a 1 Q & 7 ( ( ¤ Y £ )3 I
N ( ! #7,5 ` 3 £ O N ! 1 ^ 1 ^ ( ] 5 ` ¢ £) ¢ Q a Q & O7 £ Q& <3 £ ¤ 4 3 11.
Q& 1 1 ! 7 ( ¤ C7 ( ! 1 N ! ( V7<5 ` 12. a Q - a 1 Q & ^ 13. -21 Q 7 a &4 Q & C7(8 ! D ¢ 3h H F £ ¢ 3h G 3 H F £ ¢ 3h GIH ¤
1 7.
14.
7 £ a Q 0& 7 8 Q ( 1F
1 5P1
¢ 3h G ¤
+
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
:1 31
¢ a 1 Q a 1 0& 7(8 43 ¢ ¢ a 1 Q a 1 & 1 43 ¢ )£ ¢ a 1 Q 43 ¢ ¢ a 1 Q a
0 &
£ 1F
¢ 3h G ¤
£ Q& ¤
H
£ ¤ -21 Q&
£ ¤ 1 & 1 Q& W !£ ¢ a 1 Q & * - £ Q -
¢ a 1 Q 0&
Q& ¤
¢ a Q a 1 0&
Q& ¤
4 3 ¢ ¢ a 1 Q a 1 & Q
W¤ -
¢ #+ a 1 . Q & ) 1 £ Q& £) ¢ # a 1 Q 0&
¤
£! ¢ a 1 Q &
L
L
£ ¤ -21 Q&
£ ¤ -21 Q&
¢ # a 1 Q a 1 0& 1 £ Q& ¤
6.13. Hermite Polynomials Hn (z) 6.13.1. Series containing Hnk+m (z) and Bessel functions
1.
J L ! $ %
"
0 & a Q
a Q
5P,
W W
#$
L J 5 7 ! N 5 7 ¤ 1
/ £ 8 ¤
, '
2.
"
J L 5_7X! $ %
a a
W W #$ L ¤ Q a & J 5_7X! N 5 _ 7
& Q 1
6.13.2. Series containing products of Hnk+m (z) 1.
2.
3.
4.
" $ ! % Q & ¢ 3 -21
" 5_X7 ! % $ -21
" $&% !
5
4
2(:7
)
,
a Q a &
1
1 5 4 2(:7 , ¢ 3 -21 )
& 0
W ZZ -
"
$&% ! a 1 & 0 7 W Z Z - W - W
£ &
7(
)
7( )
£ & # /
,
=
,
=
=
£ &
£ & # /
7 @CB
7 7 = @ @CB !
7 @CB
= 7 7 ! = @ @CB
6.13.3. Series containing Hnk+m (ϕ(k, z)) 1.
2.
$ 5_7! "" $ ! %
F $ 5 7 G " £ / £ / 8 <3 $ 5_7! "" $ 5_7! % G a F 5 _ 7 $
1 5P 3
"
£ / : ¢ !
&
£ / 8 <3 ¢
!
&
7B
7B
+
:1 31 5
6.13.4. Series containing Hnk+m (ϕ(k, z)) and special functions 1.
2.
3.
7 ( 7! " ¢ $ ! % -!&4 F ¢ G F $ 5_7 G -21 O - ] ¢ 3 ( 7X! " ¢ $ 5_7X! % - F ¢ G a 1 F $ 5 7 G £ 1 O -21 D ¢ 3 5 7! 3 "" 8 " $ _ $ ! % G a / ¢ Q F F 3 " " L J !
4.
5 7 G $ _ N
£ / :& ¤
(
! H¤ W W
( 1
#$
3 ¢ ¤
5 7! 3 "" " 8 $ _ $ 5_7X! % F 3 G a / ¢ Q a 1 F $ _ 5 7 G " " ( W W L J ! ¢ 3 N
1
"" 8 " _ 5 7 ! $ 3 $ ! % PG a / ¢ Q F $ 5_7 G F " " W W L J ! ¢ 3 N V (
1
$ 5_7! 3 "" " 8 $ 5_7X! % F PG a / ¢ Q a 1 F $ 5_7 G " " W W #$ L J ! N 3
1
#$
¤
5.
#$
¤
6.
6.13.5. Series containing products of Hnk+m (ϕ(k, z)) 1.
$ ! $ ( " / / 1 43 ¢ a 1 = O''!% % @
( " 6 ' 5 1L ( " ( " ')( J _ 5 7 ! % ( ! % 5P95
¢ ¤
# . '/
Y
2.
Y
43
3
'! % @ S - " " 7 = ' % ')( $ ! % $ ! % F G ( " " " S ¢ S 2- 1 ] ( $ ( ! % 43 ¢ - -21 43 ¢ ! % £ 3 £ 3 £! 3 £ ( B >?7 ! ! $ ! $ ( " / a 1 /
a 1 1 ( " 'g5 & L ( " ( " O Q ' _ 5 7 ! @ % 43 ¢ S = ' % ')( J 5 7 ! % (* O! % " " O'65 7! % @ S -
= ' %
' ( $ ! % $ 5n ! % F G ( " "" S ¢ S -21 (* $ ( ! % £ 3 ¢ - -21 3 ¢ ! % £ 3 £ 3 )£ 3 £ ( B >?7 !
!
6.14. The Laguerre Polynomials Lλn (z) λ lk 6.14.1. Series containing Lnk+m (z) 1.
2.
3.
4.
" (P5n'65_7X! " c $ m 5 !( (P5 5_7! " (P5_7X! ( S S %
' a 0& 5: '
&c -21 &
Sc a 0 &
= '8>?7A@CB
( 5 _ 5 7 m % 5 7 7 8 ( ` $J L F G ( 5_X7 ! 1 N 1 ^ 5 _ ! = R7 @CB ('6(*'! " $ 5n'! % c - & d ¢ 3 ( 6( O'! " a Y ¢ 3 c a c a 0Q3 & 3 43 & -21 $&% c &- & = R7 @CB " (P5_7X! " c & d ¢ X & - c c £ / ¤
5P;
+
5.
6.
7.
8.
" ( P5n ! " c a 0& ¤ m !" m! ¤ m 5_7! " ( 5 7! " c a & 1 N 1 5_ 7L J m !" !" a V & c &
N m m 5 5 1 L " ( 5 7! " J
$ 5:' ¢ 3 - c -;&-21 A + -21 . $ L 0 & c a J
:1 5) ,
m 5 5 1 ¤
! ( G c F 7
=
" £ 3h& ] 9 $ % & 9. P5n'65_7 #Q( m 5n'65n $ 5n'! % a m ! " c a & d ¢ N
5 7m 9 $ % L 10. J ! (P 5_7! ' m (*' ')(* m m ! " J 1 ( m 5n' L " ( P 5n'65_7! " #& c aa 0& ' % N '65 7 # 5 7 9 $ % 11. J !
7A@CB
= @ B
¤
5_7 ¤ L
lk 6.14.2. Series containing Lλ nk+m (z) and special functions
1.
2.
3.
4.
5.
7 a 1.
¢ ! ¢ £ ¤ + c (P5_7X! " d Q c 0& 0d F G c a 1 / Q 7 ( 1 (P5_7X! " Q c & 1 N P5_! 7 5_7 L ¤ J 7 ( P5_7X! ' ( ' V7 $&% & c a & ' % N J P5_7 ! 5_7 L ¤ " W #$ Q J L L a J 5_7X! N 5 7 5_7 ¤ ( P5_ 7X! " c 0&
a Q
)£ 3P& - a Q 3 2 d ; 4 P 3 2 d Q c - 0& c W / #$ # 7 ( L c c ; 1 3 65_7! J C 7 (%6(! N g5_7#7(%6! ( ¤
5P :
# . '/
6.
j Q &
-21 + -
.
A )
4 7 (
,
5
7(
!
7B
lk 6.14.3. Series containing products of Lλ nk+m (z)
1.
2.
3.
" " a S a 3 a 9 $ % 9 $ % Q 0 & Sc Sc $&% ( P5_7X! " c Q c 0& R 5 ! !" / d ¢ 7 ( 7 ( G D (87 H c F
c 0& - c Z Z c #- - £ / £ c c /
a Q3 &-
=
£ 9 /
'I@CB
7 = @ #B # C7 !C@MB !
=
=
7 = @ @CB !
6.14.4. Series containing products of Lλ n (kx) 1.
$
!
c 43 c 1 ( " ('! ( " (P5n'65_7X! ( " 3 (P ' 5_! 7 ! ' # ! (:7! % ] (:7X! % (P5_7X! ( ( 5 J & 7L! ( ( " " " % ( (P5_7! ' S - " ( '! " ( 5:'g5 7! " " "" ¢ S ( P5_7! " ( 5 7! " 43 - -21 # (* $ ( ! % ' % ! $9% S - -21 ¢ £ £ £) £ ! 3 43 % 3 3 = >?7 ! ! ( @CB
$ ( "
lk 6.14.5. Series containing Lλ nk+m (ϕ(k, z))
1.
" " ¢ 5 7 c - 4 4& (P5_7X! $
5P<
a F¢ G c 1 3 ¢ !
&
7 B
+
2.
5 7! "" $ ) &$ %
:1 5) :
" F $ 5)7 G 1 £ / Q
(
!
&
7B
lk 6.14.6. Series containing Lλ nk+m (ϕ(k, z)) and special functions
1.
2.
3.
$ 5_7! " " ( P5_7X! "
¢ Q c F $ 5_7 G " " 7 ( D N (: 7 ! ( L3 ¢ H ¤ 1 J $ 5_7! " / " (P5_7! " d ¢) ¢ Q c F $ 5_7 G Qc& -21 ¢ 3 d ¢ R Q& - c £ / Q ¤ c $ 5_7! 3 "" 8 " ( P5_7X! " G a / ¢ Q c F $ 5_7 G F " " W #$ ¢ 3 N ( ¤ J L !
4.
5 7! 3 "" 8 " $ _ ( P5_7X! " G a / ¢ Q c F $ 5_7 G F 3 " " W J L ! N
#$
3 ¢ ¤
lk 6.14.7. Series containing products of Lλ mk+n (ϕ(k, z))
1.
2.
$ 5_7! "" a ¢ (Pn 5 '65_7X! " c Q c a F & -21 (9!' ' % & ¢ 3 02d X Q& R+ 1A $ 5_7! "" 65_7X! "
$ 5 7 G . c c -21 £ / Q
ac ¢ Q F $ 5_ 7G 5n' ( & -21 ( &' ! ' % D ¢ 3 N P ! LH 1 J # 5P L
"
"
B
B
0 ' /
6.15. The Gegenbauer Polynomials Cnλ (z) λ lk
6.15.1. Series containing Cnk+m (z) 1.
2.
3.
4.
5.
6.
7.
8.
( &! " a ¢ c & 3 - c -; c F 7 ( G & $ % = C7 (* 6( '! " c - & C7 ( 9! " 9 $ % ¢ 3 c -21 ¢ 3
c F 7 ( I5 G = ( &! " ¢ 3h
c a 0& O S P5 5 f
S 1 L " J
9! ! ( % S S $ L F m G % (P5 ! ¢ 3h
P5 1 J L J ( 5 5_7 # 5 5 Y N = P5 5 1 7 (8
1
! ( &! " ¢ 3h c a & f O S P5 5
SEa 1 1L" J S & ! ( & m % ( P5 5_7X! ¢ 5_7! % S $ G
L h 3 F P 5 J 1L ( 5 5_J 7 P5 5 5_7 Y N = P5 5 1 7(*
1
! " C7 ( &! " c - 0& ¢ 3 d2 c - - c £ ¤ " ( C7 ( &! " c - 0& 1 N 1 ! 1 ¤
" '
( X 7 ! $ 5n'! % - -; 0& '! % A / ¤
m !" m # W ¢ C7 (& ! " c - 0& 3 - N 1
1!
-Q1 5P P
7A@CB
7 @CB
7R@CB
7A@CB
#$
=
7A@CB
+
:1 !; ,
" P5_7 £ C7 (&! " c - a 0& d2 1 N 1 & ( 4 ¤ 9.
1 !
" ( 7X! ' 5n'65_7X! % - a -;&-21 0& O'Q5_7! % -21 a 10. $
1 " 7 ¤ - a -21 & 3 " 11. & / & L " 1 J
" ¤ - a -21 A 3 12. / _ 5 X 7 ! $ & L "
1 J
$ 5n' C7 (* Q(*'! " c a - & C7 ( 9! " $ L 13. J ¢ £ & 3 R c -21
6.15.2. Series containing
λ lk Cnk+m (z)
/
1.
2.
¢ !£ d ¢ a a Q c & J (&L ! 4 3
c 1
a 1
4.
5.
c & ¢ + 1A- c .
L" a a Q c a
1 1 L)" c 1
¢ + . * ;0 3 1A- c 1 ( L " 1 L " £ ¢ J ( &! J " 0 3 ¢ 3 d C7 / £ _ 3 ¢ & )£ ¢ d J 5
J
;9N N
¤
c Q3 £ £
¤
Q& ¤
1L" )£ a Q d2 J 5
1 L)" c J
" 3
and special functions
¢ !£ d2 a Q c & J (&L ! 4 3
c
3.
1 A /
/
Q& ¤
P5 1 J ( &! L c -21 F
3 ¢G ¤
& 5 1 J (& ! L c -21 F
3 ¢G ¤
c - & Q a
- c 1
C7 (9! ¤ c
- c F 8 ( 7 G ( % 1 J L
0 ' /
6.
7.
( L" &L" 1 £ ¢ J
C7 (&! J " 0 3 3 d 3 / £ P 3 ¢ & 1 )£ d2 J 5
J
& )£ ¢ d J 5
J
c 0 &
N &
c a 1 P 5
L" a Q 1L"
" / ( 5 7! 5_7X! (Q(
c - a 1 &
¤ c -21 F (:7 G
c _ 5 7 ( #$ ¤ ! 1 5_7Q( 5_7
a a Q c a & c- 1
1 #$ /& c 5_7 c a & ! ( ¤ 5n ! N & P5 1 5_7Q( 5n
(1 L" 1 L" 3 £ d ¢ J C7 (&! J 0 3 ¢ - a Q " - c - a 1 Q W " C7( &! L / - c W #$
c & ! ¤ Y c - 0& 5_7X! J & (%6( L N & 7 ( 5_7 & ( Q-Q( 1
J
(1 % L " & L " 3 £ d ¢ J ( &! "'& J 0 3 ¢ - a Q - c - a 1 Q W & Q(:7! " / - c W #$ /
c < & ! ¤ Y c - a 0& N 6( -Q1 _ 5 X 7 ! ( Q ( ( ' _ 5 7 (
&
1 & & L J
/ L d2 a c Q c & J (9! ¤
10.
9.
L" a Q a c - Q 1L"
/ 5_7X! J ( QL ( _ 5 7X!
8.
¢ - a Q
- c 1
C 7(%9! . +c a 1
( L 1 J
11.
12.
)£ J (9L ! d2 a c Q c &
13.
)£ ¢ J (&L ! d a c a 1 Q c a & 1
/
Q& ¤
;9N1
/
Q& ¤
+
14.
:1 !; 3
¢ )£ d2 J 1 L " a Q c 0 & 4 3 5 1
c
J L" 5 " 0 3 ¢ + A1 - c . J ( &! 1 L c -21 F
3 ¢G ¤
15.
16.
¢ )£ d ¢ J & L " 4 3 5 1 J L * ; 3
a a Q c a 1 0& " c 1 ¢ + 1A- c .
P5 1 J ( 9! L
c -21 F
3 ¢G ¤
( L" 1L" 1 % £ ¢ ¢ J 3 d C7(%9! J " 3h O - a + 1A- c . Q c - 0&
& 3 / 8 g+ 1A- c . - N &- c ! W ¤ - 7(1 1 1 1;- c L J ( L" &L" 1 % £ ¢ ( 9! J " ¢ 3h O - a + 1A- c . Q c - a 0& 3 d J C7%
1 #$ & &<- c 3 / 8 d2 g+ 1A- c . - N ¤ ' ( W 1 1 1;- c !
J L 1;-
17.
18.
2d 7 Y c J L
#+ a c . Q a c 1 N1 5 ! 7 c
c 0 & £ - c c #$ #$ 5_7 c ! ¤ ( 5_7X! a N a 1 1 c & & c 1 J L
λ lk 6.15.3. Series containing products of Cnk+m (z)
1.
( &$ % C7( & ! " 1 \ c - Q c - 0& c -21 N 1 7 (* C 7 (&! " 5 7X!A M E(:7! 7 ( R 8 ( 7!A M 5_7X! 7 "\ \ 7( R _ ! ;9N,
1 ( #$
! 1\ " 7 B
2.
0 ' /
£! 4 3 V d2 7<5 Y c
) * , c
£) 4 3 V d- 758 Y c a
1 ) * , c
3.
4.
)£
1L" J & ! " a c Q P5 1 7 (8 . L £ ¢ J , ( 9!
c -21 1 ] 3h
+ 1A- c *) Y F ¢ h 3 G c -21 F ¢ h 3 G ¤
¢ J && ! L " "'&
a c a 1 Q
E5 1 7,( . L £ ¢ J +
c A 1 (9!
c -21 1 ] 3 O a 1) * , ¢ ¤ Y F ¢ 3_ G G F 3
c -21
7,5n 7 (8 $ ! % d2 9! " a c Q c ) * , c ) * , #$ a a " c c 1 c 1! ( ¤ / c - & N & # P5 1 1 cL J
$ 5_7X! % d ¢ P5_7! " a + c a 1 . Q 7<5 7 (8 3 " / 8 a . ¢ , c a 1 * , g+
h 3 c 1
a ) c 1L J a 5_7 5_7 (* #$ ¤ Y N c 1 c # 5 c # P! 5_7 & & & 1
5.
)£
Y c a
1) *
λ lk 6.15.4. Series containing Cnk+m (ϕ(k, z))
1.
2.
I " ¢ $ 5_7! " ¢ ¢ C 7(%9! " c - 4& 5 7! 3 1N 1 ! " $ 5 7! "" ¢ C7% ( 9! " c - F $ 5_7 G #( Q(:7X! 3 1N 1 (
( ;9N 3
( 6 ( 1 ( ( ! &
!
Q(:7
1
!
#$
7 B
&
7B
+
3.
$ 5_7! "" C 7(%9! " c - a F $ 5_7 G
1
"
:1 !; ;
1N 1
4.
5.
!
3
1
3 ¢
E(:7 $ 5_7! " C 7(%9! " & c - * 5 $ ,
) " ('6( 1 #$ ¢ 5 7! M : ( 7X! 3 1 N 1 ( 1 ! C7 (* ]!
( $ 5n ]! "'& (:7 C7 ( &! " $ 5_7! c - a ) * 5 $ ,
1 ' ( 6( 1 #$ " P5_7X! (87 1 N 1 1 C7 (8 ! 3 ¢
! (
¢¤
a 1
&
!
!
&
7B
7B
λ lk 6.15.5. Series containing Cnk+m (ϕ(k, z)) and special functions
1.
5 7! " 3 "'& 8 " $ _ ¢ Q c - ¢ 4$ ¢ / G C 7 (9! " a ] F 3 " W \#U #$ ('6( L 1 PJ 5_ 7X! ! N ¢ ! 3 ¤ (
1
2.
5 7! 3 "" 8 " $ _ C 7 (&! " G a / ¢ Q c - F $ 5 7 F
" 6 " (87 L ]J ( 6 8 ( 7 ! !
( 1N
G !
(
3.
1
W W
#$
3 ¢ ¤
5 7! 3 "" " 8 $ _ C 7 (&! " PG a / ¢ Q c - a F $ 5_7 G F
1
" ( W W #$ " L J ! ¢ 3 N ¤ ! 1
1
;9N95
&' 3 465 78
6.16. The Jacobi Polynomials Pn(ρ, σ) (z) 6.16.1. Series containing Pm
(ρ pk, σ qk) (z) nk
" 5 g5_7 . 65_7! " j + - l & V- 1 N 1 g5_7 + a 1 . ¤ 1. !
m !" m 5 65_7 #$ . ¢ A7 @CB 65_7! " j + - l 0& - N 1 g5_7 + 1 a . = 2. ! a .
X+ 1
5 65 5 7! " 7(8 . 3. O S 5 5_7! " F G j S + a l 0& f S m 5_7X% ! ( S $ G L F J ( 5 5_7
5 g5 5_7X! 7 (8 5 5_5 7 g5 5 5_7 Y % 5_7 ! G F N
1 ! 1;-
7 (* C@ B = " E ( . . 5_7! " j + lC- - & F E ( G + a 1 F G ¤ 4. 5.
6.
7.
8.
" . ( ( ! " j + - l - &
( . - + a 1 1 N 1 J ( (! L ¤
m !" . ( ( ! " j + - l V- & F ¢ 4 ( G - N ( ( m V ( =
1 !
- ,-
: 5 ' $ . j + a - l - 0& $ L J F ¢ R M 5 7! G F ¢ R M E (:7! G j + l . ( ) ( (*'! " . . j + - l a 0& ¢ 3 j + l Q3 & 9 $ % = ;9N;
,
7A@CB
¤
7A@CB
+
:1 :1 ,
a . m ! " 1 ( m 5:'! " 5 65n'65_7! " £ ¢ 5n'65_7X! " j + a lV- & $ & % 3 h &
m ( ' ]')(* m 5_7 5 g5n'65_7 7! ' 5_ ' % &N
'65_7 5_7 1;- !
. m ! " J 1 ( m 5n' L " ( (*'! " (:7G j + a - l - &
% ( ( )(*'! " 9 $ F 10. m (*' ' ( m 5_7 V( (*' 5 65_7! ' (:7 ' %
5 g5_7X! ' F G & N 'g5 7 V( ( (* '
! 1;- 9.
6.16.2. Series containing Pm
(ρ pk, σ qk) (z) nk
1.
)£ ¢ 5 g5_7X! " a Q - 5_7X! " & & 5 g5_7X! 4 7 5_7! 5 J L ( 5n ! N m m
&
2.
( ! " )£ G a Q 3 I ( ( 2! F 7(8 4 3 5_7X ! J C 7 L ( ( m m
3.
¤
and special functions
a a a Q j + l . 0& - 1 W W a 5_7 a a & ,- #$ ¤
m 5_7 5_7 5 ! ( m 5n
. + - l & j Q - - W #$ # 7 ( ! N & m 5_1;7- V ( ( #! 7,( -Q 1 ( m ¤
( ! " . )£ 3 I ( ( ! " F (:7 G - Q j + - l 0& 1;- ! ¤
G F N ( L 1 1 ( ( 1; - J
5 g5n'65_7X! " 7 (8
. C7 ( &! " G c - Q j + a l & F
5n'65_7 5 g5n'65_7 #$ 7! ' 7<5 W 5_ ; & 2 1 ' % F G N&
1 5_7 + ,-Q1 . ! a 1
= 7(* @CB
4.
¤
;9N :
&' 3 465 78
5 g5n'65_7X! " 7 (8 . C 7 (&! " G c - a Q j + a l 0& 5. F
1 5_7 5n'65_7 7X! ' 7<5 £ d 5_ ' % F G - - -;&-21 & N
& 5_7 !
5 g5n'65_7 #$ W
. + ,a -Q1 1 7 (* =
@CB
(ρ pk, σ qk) (z) nk
6.16.3. Series containing products of Pm
5:7! " ¢ !£ ¢ $9% 5n 7X5 ! Q 3 " 5n7! "
& & 5 65_7! L 4 7 . . Y a Q a a a Q j + l 0& j + l 43& m 5_7X! 5 J ( m 5: O! - 1 a a a 5 7 a _ 5 7 a a & ! ( ¤ Y N 5_1 7 5_ 7 65_ 7 5 6
m
5_7 5 ( m 5n
1.
(ρ pk, σ qk)
6.16.4. Series containing Pm nk a 4-4# $ ¢ =
(ϕ(k, z))
1 " . $ 5 7! ( ( ! " F 3 T G j + - l - ¢ 7X! " 5 ] g5_ 5_7X! . $ 5_7! "" 5_7! " j + l V- F ¢ $ 5_7 G 9 $ % "
5 ! 3
1.
2.
6.16.5. Series containing Pm functions
¢ && ¢ 3
(ρ pk, σ qk) (ϕ(k, nk
1.
$ 5_7! " 3 '" & 8 " ( ( ! " ]F 3 G a 5 g5 7! # 65_7! J
( 8 ( 7! (
¤ 1 N 1 ( ( (87
5 (
¢ 3 1 N 1 ! ` ¤ ^ z)) and special
. / ¢ Q j + - lV- 4 ¢ 463 ¢ " ( (:7 ( W \#U #$ L ¢ ¤
! 3 1 N ( ( ! (87 ;9N<
+
2.
:1 !<
5 7! 3 "" 8 " . $ _ 65_7! " G a / ¢ Q j + - l F $ 5_7 3 ¢ G F 3 " " 5 ( L J 5
¢ 3 1 N ! 2! ! $ 5_7! " 3 "'& 8 " g5 7! " G F 5 5 7 Y Q3 3 ¢ a / ¢ Q j + - l . F ( $$ (87G " " ( (:7 ( W L ! J 5_7! ! 1N
. $ 5_7! " 3 "'& 8 " ( ( ! " G a / ¢ Q j + - lV- 4 ¢ 463 F " ( (:7 W \#U #$ 5 g5_7! L ] g5_7X! J ! N1 ( ( )! (:7 . $ 5_7! 3 "" 8 " 65_7! " G a / ¢ Q j + - l F $ 5_7 3 ¢ G F " " 5 W L J
! 5 ! ! 1N
$ 5_7! " 3 "'& 8 " g5 7! " G 0Q3 3 ¢ a / ¢ Q ] F 3 " " ( (:7 5 _ 5 7 $ ! Y j + - l . F ( $ (:7 G J 5_L 7X! ! ¢ 3 N 1
W
¤
3.
3 ¢ ¤
4.
¢
3 ¢ ¤
5.
3 ¢ ¤
6.
W
¤
6.17. The Generalized Hypergeometric Function p Fq ((ap ); (bq ); z) 6.17.1. Series containing p Fq ((ap (k)); (bq (k)); z) 1.
2.
" m ! " 1 N 1 5! $ & % J " ( ! 5_7X! % 1 N 1 & $ % $ J
$
& f
¤
¢ £ £ !5 L $ n & 3
£ 1 & £ £ 1 & H & <3 ;9N L
£ & H £ & ¤ 1
! ( ! ( m n 5 $ m " m " J 1L " J1L " £ ¢ ! " (87 ! " & 4 1 N 1 5 $! L $ & % 4 3 3. J m
N1 1 ^ ! ` ¤ ( 5 ( 5 m 5n $ Jm 1L " J m 1L " J 1L " J 1L " £ ¢
$&% ! " (:7! "
& 1 N 1 5 $! L 4. 4 3 J ( 5 N m 1 N m 1 ¤ 1 1 ! 1 1 !
!" m 5 $! ! 5 $ L = 7A@CB 5. 1 f $&% " N 1 J ( $ $ 5 7 ] ( 7(* ¤ " !7 L N 7 V 7 7 9 $ % L 6. N N 1 1J 1 1J 7! L & J ! ( $ #7 7 #$ " 1 ! 7 3 1 / L / 7. $&% ! & N $ 5_ 7 &
£ ¤ ¢ / V L1 / 0 ( $ ( $ ( #$ "'& 1 m ! m 5_7! & N
= @ B 8. ] ! 0f
& $&% $ 1 ( $ m 5 7
#$ " ( ] ! 1! fI I+ 1A- . H £ ¤ m ! " 1 N m 5 $ & $ % 9. ! - &4 / &
$ 5 #$ $ 1! 5 7! % 1 N $ 5 10. $ & & 1
7 & ( 5n ! ( K(* ! ¤ 3 5n (*
*
11.
12.
m! ( ! " _ 5 5 7 L 0f ¢ 4 -R £ / ¤ $ N ! 9 $ % 1 J 7 m _ #$ ( ! " 1! 7 0fI / I + 1A- . H £ / ¤ m ! " 1 N $ 5 m 9 $ % -!&4
&
;9N P
+
13.
$ ( 7 $ 8 $ ! % 1 N $ 5_! 7 & 1
£ (! /
14.
" ! $ % 1 N
1 2(
15.
16.
17.
18.
19.
20.
21.
22.
23.
1
1
:1 !<
£ / <3 *
/
£ / <3 £ ¤
$ $ 5_! 7 1
D / £ / 3 £ " / £ / 3 ¢ H ¤ $ 7 ! " $ ! % 1 N $ 5_! 7 1 £ / ¤
$ 7 $ (87 $ ! % #& 1 N $ 5_! 7 & £ / ¤
#$ $ 5 " 7 & ( K( ! ( 5n 2! ¤ 1! $ 5 7! % 1 N $ 5 K(* 5n & 1
$ 5 #$ ! " 7 1! 7 $ 5 7! % 1 N $ 5 / ¤ & 1
" ! 7 ]! ¤ $9% $ 5_7X! % 1 N 5_7! a &
7 ( ]! " 7 $9% $ 5n ! % 1 N a & ! 5n ¢ 3 / ¤
( #$ ( ! " 1 ! ¢ 3 / H £ / ¤ $9% 1 L " 1 N 1 $ 5 1 J
7 ( ( 7 ! " 7 ! ] 1 $&% ! 1 N a 1 a a & 5_7 7 / £ 1 ¤ 7 ( 7 ! " ¤ $9% $ 5: O! % 1 N a & a ! a a O
;N
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
( ! " $ 5_7X! % N 7 $ !5: L £ $ V ¢ £ & <3 £ 1 £ & 9 $ % J £ ¤ £ £ £ £ ] 1 & H & <3 & H1 & " $ 5 $ 5_7 #$ JL 1 7 " 0/ ¤ ! N & L " $ 5 & $ 5n J
( ! " 7 #7 $ 5_7X! % N & $ ! 5n L O £ / 9 $ % J 7 7 3 F C G £ / ¤ #$ ( ! " 1 #7 ! 7 £ / <3 £ / 0 ¤ $ 5_7X! % N & $ 5: 9 $ %
&
" #$ ( ! 1 #7 ! 5 7! % $ 5 ! N & $ 5 ] $ &
7 £ £ ¤ / <3 / H-21 / ! " m #7 ! (:7X! " N & m 5 -Q1 m 5 m (:7 $9% m
£ ¤ £ £ &- fg3 4 &4 - -!& / 7 #7 ]! " $ 5n ! % N & a & ! 5n 9 $ %
7 £ ¤ / C / / 7 #7 ( ( ]! " 5n ! % N & a & ! 5n $9% $
7 7 V/ 3 £ £ / C / ¤ 7 #7 ( ]! " 7 Z1 ! ! ¤ ! 5n ! % N & a & 5n 3
7 ( $9% $
" #$ ( ] ! 1 #7 ! 7 V/ H / N !
9 $ % a _ 5 7
& 1 1 &
¢ £ ¤ V/ 0V 3 H1 / 0 ;
+
34.
( ] ! " 5_X7 ! % N & $9% $
35.
36.
$ 9 $ % $9% $
#$ 1 V 7 ! 5_ 7 a & &
a ! 5n #7 &
m #7 a 5n! ( &
]! " 5n ! % N & ]! " 5n ! % N &
37.
38.
39.
40.
C( ] ! " _ 5 7X! " N & $9% m
42.
H
/ ¤
&
! ¤
m (* m m (:7! / " . ¤ m (:7 3hfI I+ -!& A1 - /
#$ 1 # 7 ! a 5 m 5 m _ 5 7 & 1
£ - £ fg3 ¢ - £ f ¢ Z 7 3
-21 H V/ 8 = m h7 @CB #$ ]! " 1 V7 ! 5n ! % N & a 5n $9% $ &
¤ V/ / <3 V/
#$ ( ! " 1 7 ! $ 58 ! % N & a 5* 9 $ % &
" #$ V 7 ] ! 1 ! 5n ! % N & a 5n $9% $ &
D / !
! " 5 7 ! " & N $9% m
#$ ! a a 1 m 5 7 a a a a ! m ( 5_7
1
£
- 0fg3 ;!,
H
V 1 V/ 3
7 C7 ! " _ 5 X 7 ! % N a a $9% $
&
41.
:1 !<
¢ ¤ 1 V/ 3 H
V/ ¤
7 7 #/ ¤ #$
¢ 4I+ - . / ¤ -
"
#$ m m 5 & #7 ! ( ! 5n ! " & N a 5 m a 5 m m 5 m 5n 43. 9$ % m
& 1 " " " "
_ 5 7 ( m m m 5_7 £ f a 1 V/ m 5_7 1 N J m 5n ! m 5n L ¤ " #$ # 7 7 ( 7 7 ( ] !
$ 5n ! % & N a & 5n& ! V/ <3 1 N ! ¤ 9 $ % 44. & & &
& & " #$ 7 ]! & & V/ ¤ $ 5_7X! % & N 5_7 a ! 9 $ % 45. & 1 1
" 7 ( V < 7 5 ] ! m m V7 $ 5n ! % & N a & 5n ] ( m ! ] 5 m 9 $ % 46.
7 ( Z m ! ! m m m ! / 3
¤ 7( m " #$ $ 5 $ 5_7V7 ! 1 & ! 7 £ ¤ 5_7X! % $ 5_7X! N $ 5 5_7 a " / 47. $ & 1 &
m !" ! ! ( & $ " " m m !95 $ 9 ! 5 $ ` V & a 48. N ! " 1 ^ m 5 5 1 L " ! J
m ! m a N a ¤
1 ! m 5 5 1 !
(')( $ & m ! '65 m ! " a N a 49. 1 1 ^ m ! ! ` $&% )" " " ! m !95 $ ¢ 3 -;&- ' $,L (8 7! ! m " N 7A@CB !&5 ! $ -Q1 = !" m " J m ! ¢ m ! E ( C7( ! " !( $ ` ! ! 2 1 ! & $ % a 3 a 50. N N ` 1^ 1^ = 7R@CB (')( $ m ! C7( m ! " a N a & $ % 51. m 1 1 ^ ( $ ! ! ` " " ! ¢ 3 -21 ' $L 2(: 7X! ! m " N m !95 !95 $ $ ` = 7R@CB !" m " J ^ ! ;3
+
52.
53.
m !95 $ ( ]! " m ! " ! " ! " N !95 $! ^ m !95 $ ( ! " m ! " $ 5 ! ! " N !95 9 $ % ^ ( 1L" J
$9% m 5 1 L" 3& J 7 £ fg3 ¢ m
`
:1 !<
a 1 N a 1 ^
m ! (:7 ¤ ! ! `
7 m ! #7 ¤ ! 7 ` $! ` a 1 N a 1 ^ ! 5_
m !" m !95 $ !95 ! 54. N !" ^ $ ` m ! m m ! m ! a 1 N a 1 ! m ( 1 a 1 N a 1 ! m 5! 1
("'& ( m !" 9 ! 5 $ m m $ 5 ! % ! " N !95 ! $ ` 55. ^ " ( C7,(: ! ! ( S ¢ $ L N m !I( !I! ( $ 5 m ! 43 ¢ S + a a 1 . % 3 C7,(8 m !0! ( ^ J m !95 $ " . m !" + M l 1 _ 5 X 7 ! j $ % 56.
! " N ^ !95 $! ` a N a m ! ! 5: ! 1 1^ m !95 $ V7 m ! m 5_7 ¤ m !" m !" !95 $ ! ` a 1 N ! & $ % !" a 1 N 57. ^ ^ ! ` m !95 $ m m !" S !95 ! 58. f O ! " a 1 N ^ $ ` S m !95 m 5 5_7 m I m ! S !95 ` G J $ L F ! a 1 N ^ ! = " m !" m !95 $ m ! " ! " N a 1 !95 $ ! ` 9 $ % 59. ^ m ! a a -Q1 ! a N a ! m 5 (:7
m
& ;5
¤
` ¤
` ¤
7R@CB
¤
60.
m !95 $ ( ]! " m ! " ! " ! " N a 1 !95 $ ! ` 9 $ % ^
61.
N a &
m !95 $ m ( ! " m ! " $ 9 % ! " a 1 N a 1 !95 $ ! ` ^
J
J
W L a 1 L ( \ U #$ ! ¤ L J a 1 L a 1 J
m ! ( m ( ¤ ! ! ` a 1 N a 1 ^ m !" m !95 $ m m !" ! " 43& ! " a 1 N a 1 !95 $ ! ` 9 $ % 62. ^ W # . . a ( m m \
U $ ! ¤ a N a + +. . 1
& + + a 1 a 1
m !" 9 ! 5 $ ! ! m m m " " ! " a N a 1 !95 $ m 5 ! 5 63. 5 5 9 $ % m 1 1L" J
m ! m m 5 a N a ¤ !
! m 5 5 1 m 5n &
m !" !" 1 ( m ( L " 9 ! 5 ( $ $ m m J % C7 ( m ! " #& ! " a N 9 ! 5 ` $ & $ 64. 1 ^ ! a N a m ! m 5n! # 7 #( 7( m (* ` ¤ m!
1^ !" $ 5 m & m ! 5 ! 9 $ % $ $ 65. m a 1 N a 1 ^ 5 m 5_7 ! ! ` m m ! ! 0f ¢ 3O a N a 7A@CB ( _ 5 7 ! = ` m 1 1^ " ! m !&5 $ 1 V7 #$ m " ( ] ! ! 66. $9% ! ! " a N a & !95 $ a 1 a &
#$ m ! m !95_7#7 ( ( 7 7 m 1 a N a ¤ ! ! 3 a 1 N a !95_7 & 1 ! #7 &
;!;
+
67.
68.
:1 !<
m !95 $ #7 ( ]! " m ! " m ! " ! " a 1 N a !95 $ 5 m ! a 1 5 m 9 $ %
m ! m ( ( #$ 1! ¤ a N a
! 5 m 1 ] m (:7 1
m !&5 $ #7 ( ]! " m ! " 5n ! % ! " a 1 N a !95 $ a & ! 5n $9% $ (:7X!
7 ! : ( 7 ( ¤ m ¢ 3 N a ! ! : ( 7 # 7 ` m (:7X! 1^
( ! " m ! " _ 5 7X! " ! " a N a 69. $9% m ]! " m ! " 5n ! 70. $9% $ % ! " a N a & (:7X! * ( m (:7X!V m m (:7X! N a 1 ^
71.
72.
73.
m &! 5 $ m 5_7 m ¢!¤ ! & !95 $ a 1 5 m 5 m 5_7 m
m !95 $ m #7 !95 $ a & 5n! ( m
m !(:7 m !(:7 !(:7! V7 ` 3 N a 1 !(:7 ! ( m ` ¤ ^
( ] ! " m ! " 5n ! % ! " a N a & $9% $ m ( ]! " 5_7X! % a 1 N a !95 $9% $
m 9! 5 $ 7 ¤ !95 $ a & ! 5n # 7
!95 $ #7 $ 5_7 ! a & #$
! ( m 1! ¤ a N a 1 ! # 7 &
m !" " ( 7 ] ! 5_7X! % m (:7! " m ! " ! " $9% $ #$ m !95 $ V7 Y a N a !95 a a a & ! a a
$
1 & 1
m ! ( ¤ N a ! & ! m (:7 a 1
;:
74.
75.
76.
77.
78.
C 7 ]! " $9% ! ! " Y a N a
7 a N a
m !" !" !95 $ ! m ! #7 ( m
m a
!95 $ m V7 a a !a 1 a 1 7 #$
a ! ( m 1
1 a #7 ( m 7
m ! 7 N a & ^ ! ! #7 ` ¤
! " m ! " (:7X! " ! " m 9! 5 $ #7 m ! a & N a !95 $ Q - 1 5 m 5 m m (:7 m ( (:7
m ! m ( (* ( ¤ ! N a 1 ^ ! m (* m ( (:7 ` ( ! " m ! " 5 ! " ! " $9% m !&5 $ 5n #7 Y a N a !95 $ m a 5 m 5 m 5n ! 5_7 5 m m m & & m
m ! m 5n ( ¤ ! a 1 N a ^ ! m 5 m 5n ` ! " m ! " 5 7! " ! " $9% m #$ m !&5 $ m 5_7 Y a N a !95 a a a a ! ( 5_7 $ &
m 1
! m N a ! ( ! 5_7 ` ¤ 1^ m ( ! " m ! " 5_7X! " ! " $9% m #$ m !95 $ m m 5_7 m ( Y a N a !95 a 5 1 a 5 1 ]! 5 $
m & m m 1 1 m
m ! #7 ( ¤ a N a ! 1 ! & m 5 1
( m 9 $ % Y
;!<
+
79.
80.
81.
82.
83.
84.
85.
m 9! 5 $ 1 #7 & ( ] ! " m ! " $ 5_X7 ! % ! " a N a !95 $ a & & a ! 9 $ %
& 1
& a N a 1
( ]! " m ! " 5 ! $9% $ m ! " N
:1 !< #$
m & ! #7 ( ¤ ! !
m 9! 5 $ m !95 $ 5 1 #$ ! !95 $ & !95 $ 5 1 ! !95 m m 7 a N a m 1 1 ! !95
& &
#$ 1 # 7 ! ¤ 1 m 5_7
m !" ! 9 ! 5 n 5 $ $ m " m m ! !95 ` $9% " 4 3& ! " a 1 N ^ $ ! a N a m ! m ! m ( ( 5_7 `
1^ ! m !95 $ m 5n $ m !" J L " !" !95 `
% ! " 3#& ! " a 1 N & $ ^ $ ! R. R. a a ( #$ a N a + . + . 1 1 W
1 + + a 1 5 1 ! \
! " m !" !" m !95 $ m 5n $ m ! " m 5 ( 5_7! " 3P& ! " a 1 N !95 ` 9 $ % ^ $! ( a N a m ! ! m m 5 ( ( 5_75_ 7 ` m
1^ ! m !&5 $ m 5n $ m m !" m !" !95 ` & $ % !" a N a 1 P 3 & ^ $ m!
.
.
. #$ + + a 1 + a a 1 a N a & & 1 + . & + . a 1 & + . a & m 5 1 ! ( \
& & & m !95 $ m 5n $ m !" m !" ! " 43& ! " a 1 N !95 ` 9 $ % ^ $ ! a N a m ! m ! m ( ( 5_7 `
1^ ! ;L
¤
¤
¤
¤
¤
m !95 $ ( ]! " m ! " m ! " ! " N a 1 !95 $ m ! 5n $ ` 86. ^ 1 m !95_7 a m 1! 3 5_7! a N a m m 1 !95_7 a & m 5 7
$ 5 m (87!& m (:7X! " ! " ! " ( ! ( ! " m (:7! "'& 87. $&% m " m m ! !95 ! ( ( Y 43& ! " N a !9m 5 $ $m ! 5n $ ` a N a m ! m m ( m !( " 1^ 1 ^ m (:7! " $ 5 m (:7 m !95 $ m !" $ 5 m (* !A $ 5 m ! $&% m ! " 3P& ! " N a 1 !95 $ m ! 5n $ 88. ^ m (:(*7 O! a N a !m ! V 7 ! 5_7 m m 1
m !" 9 ! 5 $ m $ 5 m (:7X!A m ( ! " m (:7X! "'& ! " 3P& ! " N a 1 !95 $ m ! 5n $ ` 89. ^ 7 a N a m !! (: : ( m 7 ! 1 ^ m !" m !95 $ m !" ! 9 $ % a 90. " 3P& ! " N 1 ^ !95 $ ! 5n $ ` ! m - - a 1! a N a
& ! a 1 ( m
m !" : ( X 7 ! ( ! 9 ! 5 $ m m " " m m ! " ! " 3P& ! " N a 1 !95 $ m ! 5n $ ` & $ % 91. ^ ! m 1;- 5 ! a N a ! a 1 1
m (:7! " $ 5 m (:7 m !95 $ m !" 5 !A 5 (* O! $&% m ! " 3P& ! " N a 1 !95 $ m ! 5n $ 92. $ m $ m ^ m (:(*7 O! a N a !m ! V 7 ! 5_7 m m 1
#$
¤
;P
` ¤ `
¤
` ¤
#$
¤
#$
¤
`
¤
+
93.
94.
95.
96.
97.
98.
:1 !<
m 9! 5 $ $ 5 !A ! " m ! " m ! " ! " 3P& N a 1 !95 $ m ! 5n $ ` $ 9 % ^ ! m - -Q1 - ! #$ a N a
& ! a 1 m (
m !95 $ $ 5 m (:7X!A m (:7X! " m ( ! " m ! " $9% m ! " ! " ! " N a 1 ^ !95 $ m ! 5: $ ` 0fg3 ¢ N a m ! ! ! ` 1^ C7( m ! " m !95 $ m !" $ ! % m ! " ! " ! " N a & !95 $ 5n $ ! ] $ 5_7 ` ^ m ! a -Q1 a #$ 7 m ! 7 a N a ! ! ! #7 N a &
! m a 1 m 5 (:7 ^ `
" m !" ( ] ! m !95 $ $ ! % ! " ! " N a & !95 $ 5n $ ! $ 5n (:7 ` ^ #$ m ! N a ! & ! a 1 (:7
" ! " ! m ( O 9 ! n 5 m $ m m JL m !" !" N !95: $! ` a 1 N a 1 ! m 1 (8! 7 ¤ 9 $ %
^ ( ! " m ! " m !95n $ ! ` $ 5_7X! % ! " N a 1 !95: $ $ 5: 9 $ % ^ m R. a R. 5_7 ( #$ ¢ £ N a + . 1 + . 1 !
1 + a 1 + 5_7#7 &
. a . 5_7 ( #$ + 1 + . a . 5_! 7 3 N a 1 + 1 +
m !" ! ! 9 ! n 5 $ m " " m m 5 $ 9 ! n 5 ` a $ 3 N !" 1 ^ , $9% m 5 5 1 L " ) ! J
m 5 a N a m ! ! m 5n
1^ m ! `
¤
¤
¤
¤
99.
; ,9N
¤
¤
m !" m !" m !95: $ m 5 $
G a 100. F N 1 ^ !95: $ ! `
$9% ! " m ( 5 & L " ! " J
m ! m ( m ( 5_7 1 @CB a N a
! (:7 m ( 5n ! = & m !" m !95n $ " $ 5n ! % ! " ! " N a & !95n $ 5 $ ! $ 5 (* ` 101. ^ (:7X! #$ m !(:7 - - & ( O A ! 8 ( 7 ! ! m (87! a N a !(87 (:7 -Q1 ( (* 1
# $ m !(87 3 N a & !(:7 (:7 ! (* ¤ -Q1
$ 5 m ! ! " m !95 $ L " m m ! "'J & & 3P & ! " N a 1 !95 $ m
5 ! $ 5_7 ` 9 $ % 102. ^ ! m ! & #$ ¤ a N a 1 ! & a 1
6.17.2. Series containing p Fq ((ap (k)); (bq (k)); z) and trigonometric functions
$ ! m ! 7 65 $ ! ) ( $ ! N a ! 65 $ ! ( $ ` $ 1. ^ 1 C 7 ( ! ! N a m ! ! ! ` =
^ $ ! m ! ( 7! " 65 $ ! )( $ ! N a ! 65 $ ! ( $ $ 2. ^ 1 3 ! N a m ! ! ! ` =
^ $ 5_7X! @ ( 7! " $ 5_7 65h $ 5_=7! ! )( $ 5_7X! ! 3. ! Y N a ! 65h $ 5_m 7! ! )( $ 5_7X! ` ! N a
^ = 7
7
!
7A@ B
`
!
7 X@ B
!
; ,
m ! !
( ^ 7 ! 7 ` @ B
+
:1 !< ,
$ 5_7X! @ ( 7 ! " 65h $ 5 = 7! ! )( $ 5_7X! ! 4. _ 5 7 X ! $ ! ! Y N a ! 65h $ 5_m 7! ! )( $ 5_X7 ! ` ! N a m ! ! `
^
^ 7 ( 7 7 @ B = ! $ 5 7! @ ( 7! " $ (:7X!A $ 5 ! 65h $ 5_= 7X! ! )( $ 5 7! ! 5. ! Y N a ! 65h $ 5_m 7X! ! )( $ 5_7X! `
^ ! m !! 3 65n ! )(* ! N a ! 6 5n )(* `
^ ( 7 7 @ B = ! $ 5_7X! @ 7 $ (:7X!A $ 5 ! 65h $ 5_= 7X! ! )( $ 5 7! ! 6. ! Y N a ! 65h $ 5_m 7X! ! )( $ 5_7X! `
^ ! m !! 3 65n ! )(* ! N a ! 6 5n )(* `
^ = 7 ! 7 X@CB $ 5 7! @ $ 5_7X! $ (:7X!A $ 5 ! 65h $ 5_= 7X! ! )( $ 5 7! ! 7. ! Y N a ! 65h $ 5_m 7X! ! )( $ 5_7X! `
^ ! m ! 65n ! T(* ! N a ! g5: ! ( ` ^ = 7 ! 7 X@CB $ 5_7X! @ ( 7X! " $ 5_7! $ (:7X!A $ 5 ! 65h $ 5_= 7X! ! )( $ 5 7! ! 8. ! Y N a ! 65h $ 5_m 7X! ! )( $ 5_7X! `
^ ! m !! 3 65n ! )(* ! N a ! 6 5n )(* `
^ 7 ( 7 7 @ B = !
; , ,
9.
$ m 1
( 7! " $ m ( 65 1 3 m 6 5 J
1
12.
1
13.
! ( $ ` $ ! ) ! ! )(
=
!
7A@CB
$ ! m ! ( $ ! N a ! 65 $ ! )( $ ` $ ! ) ^ m ! N a ! 65 ! )( )( L L J 7 m ! ( 7 ! 7A@ B ! N a ! ! ` =
^ $ ! m ! $ $ m ( ! ( $ ! g5 $ ! N a ^ ! 65 $ ! ( $ ` + 1;- . m ! ! a N ! 6 m 65 L )( L
5 )( J J = 7 ! 7A@CB $ ! ( 7! " $ ! $ m ( )( $ ! 65 $ ! N a ! 65 m $ ! )( $ `
^ m ! 3 ! a N m g5 L ) ( L
! g5 ( J J = ! 7 X@CB $ 5_7X! ! ( 7X! " = $ 5_7X! m ( @ 65h $ 5_= 7X! ! T( $ 5_7! ! ! Y N a ! 65h $ 5_m 7X! ! )( $ 5_7X! `
^ m ! ! m 65 L ) ( L N a ! 65 )( J J ( 7 7 @ B = !
11.
$ ! 7 ( ! 65 $ ! )( $ ! N a ! 65 m
^ %
. m + 1;- N a 3 m 65 )( ! 6
5 L L J J7 m ! ! N a ! ! ` ^
10.
; ,93
+
$ 5 7! @ ( X7 ! " $ 5_7X! 65h $ 5_= 7X! ! T( $ 5_7! ! _ 5 7 X ! ( @ $ m = ! Y N a ! 65h $ 5_m 7X! ! )( $ 5_7X! `
^ m ! ! a N
! 6 m 65 L )( L
5 )( J J = 7 ! ( 7
14.
:1 !< 3
7 @ B
6.17.3. Series containing p Fq ((ap (k)); (bq (k)); z) and special functions 1.
m !" m !95 $ ( ]! " !&5 ! $ & % fI ! " N ^ $ ` m #7 ¤ 0fI 3 a N a m 9! !95_5_77 #7 5_ m ^ m 7! ] `
2.
1
( ]! " m !"
! If m " N m !9!&5 5 $ $! ` !" ^ m m !95_7 m ! 7 ` 3 m f fI a 1 N a 1 ^ !95_7 m 5_
m !95_7 m m a N a ^ !95_7 m 5_7 m! 5 ! £ ) fI m " N m !9!95 5 $ $! ` !" ^ m m !95 7 #7 #7 7 ! fI 3h m a N a !95_7 5_7
m !95_7V7#7 7 m 5_7 a N a !95 7 a!
m !" m !&5 $ m ! ¤ ! !
! " N a 1 !95 $ & N a 1 ! 1
3.
( ]! " &$ %
4.
7` ¤
C 7 ! " ! $ %
; ,!5
¤ &
5.
m !95 $ £ ¢ C 7 ! " m ! " ! ! 3 $ % ! " N a 1 9! 5 $ 1
m #
m !95_7 N a 1 !95_7! & ¤
6.
7.
m !" m !95 $ " & $ % Q ! " a N a 1 1 ^ !95 m !" m " j 1 L " Q ! " N a 1 !25 J
#7 $ ] ! `
N ^
!25 $ $ V $ ! 5 &
m ! ¤ ! ! `
m ! N a 1 !! 1 ¤
m !" m 9! 5 $ " ! j 8. 1 L '" & a 1 Q ! " N a 1 !95 $ $ 5 J
£ N a m ! ! 1 ! &
m !" " 9 ! 5 $ m $ ! % Q ! " N a 1 !95 $ ] $! 5_7 ` 9. ^ 7 N a m ! ! 7 N a m !! ! #7 1 ! 1 1^
m !" " m !95 $ $ 5 7! % a 1 Q ! " N a 1 !95 $ $! 5n ` 10. ^ N a m ! ! 1 ! &
m !" " 2 ! 5 $ m $ ! % Q ! " N a 1 !25 $ # $! 5h ` 11. ^ N a m !! 1 ! 1
; , ;
¤
` ¤
¤
¤
+
12.
13.
14.
15.
:1 !< 3
m !" m 9! 5 $ " $ 5 7 ! % a 1 Q ! " N a 1 !95 $ ] $! 5 ` ^ £ N a m ! ! 1 ! &
m !" " ( ] ! 9 ! 5 ! $ m m ( 5_7X! " c Q ! " a 1 N !&5 $! ` N a 1 ! # ! 5 7 ^ ^ m 5 $! !" ( m !" ! " ( 5 7! " c Q N 1 5 $ L J ( m ¢ 3h& - N ! P5_ 7 -Q1 ` = ,
^ m !" m !95 #$ " ! ( 9! " c Q ! " N a
9 ! 5 # 5 $ 5_7 1 !C@ m !95 1 N a m ! ! £ Q 1 = !95 1 ! 1 = m !C@ 1
!9 5 m 1 ! Y N a 1 !95 1 &
m !" m !95 $ " !95 # 5n! $ 5_7 ` ( 9! " c Q ! " N a 1 ^ $ N a m ! ! 1 ! 1
m !" " 9 ! 5 $ m ( 9! "'& c a Q ! " N a 1 !95 $ P5n! $ 5n `
1 ^ £ N a m ! ! 1 ! &
" m !" m !95 $ J L C7 ( &! " c - Q ! " N !95 ! ^ $ ` W . . a \
U W + + 1 ! N a ( 1 + . + . a 1 #7
¤ ` ¤
7R@CB
16.
17.
18.
; ,9:
#$
¤
¤
#$
¤
¤
19.
20.
21.
m 5 $! !" ( m !" ! " C7 (%9! " c - Q N 1 5 $ L
J ( mW ¢ 3h& - N & 1 !
,-61 m 5 $! !" ( m !" ! " C7% ( 9! " c - a Q N 1 5 $ L
1 J ( # 5 7 m W £ d T ¢ 3h& - N &
& !
,-Q1 . m 5 $! !" ( m !" ! 5_7X! " j + l V- Q N 1 5 $ L " J g5 7 ¢ 3h& - N ( m 5
5_7 + -Q1 . . &
!
X+ 1;- p Fq ((ap (k));
6.17.4. Series containing products of
( ]! " ! " $&% ! " a 1 N
1.
#$
#$
#$
=
7R@CB
=
7A@CB
=
7A@CB
(bq (k)); z)
( $ & m ! !95 $ ! ` N !95 ! $ ` ^ ^ ! a N a m ! ! ! ! ! ` ¤ ^ " m ! " C7 ( ! " J C7( L m ! " ! " C7( ! " ('65 $ & m !95 $ ! a N a 1 1 ^ $ 5 m 5_7 !95 $ ` ( $ m ( $ !( $ a N 4a 1 ^ 7 ( m (* $ !( ! $ ` & a 4a N a a (' m ! #7( ! ! ( 7! 3 8 ¤ 7 ( m ! #7 ( ! 1 1
2.
¢ + a 4 a 1 . ' $ L 4 3 J Y Y
6.17.5. Series containing p Fp+1 ((ap ); (bp+1 ); ϕ(k, x)) a
1 = 3 f 1 1 ! ( 7 $ m ¢ N a ! !& ! ` 3 4 3 1^ 1
4-4# $
1.
; , <
7 !
7 !
B
+
2.
:1 !< ;
7 m ! ( $ ( 7! ' & $ ' N a ! 1 ^ !& ! ` O'65_7X! % 1 a O6 ' 5_7 ( $ £ Y 1 GIH $ L &- DXF J m !95n')( $ @ !& !C@ Y = !C@ !95n')= ( $ @ =m = !& (')(:7 ! m ? ')(* 5h C7 7! ! _ O'65_7 !
3.
7 m ! ( $ $ m N a ! 1 ^ !& ! ` 7 #fI m #fI
m
!
W m ! W #$ N a ! !& ! 1 m !95 V7 W = m C! @ !& !95 1 1 ! W a N a
!& !95 1 & m !95 1 = !& !C@ 1
7 (
#$
!
5.
B
!
B
m ! ( $ ( 7! ' & ! ] O'! % &-21
1 ^ !& ! ` O' £ &- ')( $ 7 5_ G $ L L
F J J m !95 X@ !& !C@ Y = !C@ !9= 5 X@ =m = !& ( ' (87 h7 (* '
( 7' ! " N a $ 1 Y
4.
( 7! " m ! ( $ $ m N a ! 1 ^ !& ! ` 1 7 #fI m #fI
; ,9L
W m ! W N a ! !& ! 1 7 ( ! !
!
B
#$
B
m ! ( $ 7 7 ( 7! " ! ! ` 3 N a 1 $ (:7 N a 1 ^ !&
m m !&5_7 ( 3 !& N a 1 ^ !& !95! 7 ` 7
6.
m ! ( ! ! ^ !& ` !
(
!
B
7.
m ! ( $ $ ( " N a ! 1 ^ !& ! ` 1 S - " m ! " $9% !& ! " 3 S - -21 ( ! ¢ ! % 3 ( _ ( >?7
8.
$
!
!
3 ¢
!
¢ S
!
( " S ] ! (87 ! % ( " "" -21 ] (* $ (
£ 3 £ 3 £! 3 £ !
!%
m
!
1 !
m ! (E $ 5 m ! ( 7 ! " ! !& ! N a ` 1^ $ 1 m ! ( m m 3 7 N a ! 1 ^ !& ! `
; ,9P
!
m 9! 5 7 ( m N a 1 ^ !& &! ! 5_7 `
( 7 !
( !
#$
m ! (E $ 5 m ! 7 m ! ( m ¢ ! !& ! ` 3 N a 1 ^ ! !& ! ` 4 3 N a 1 ^ h7 h7 1 !
10.
(
m ! ( m m ! K( $ 5 m ! 7 ! !& ! N a ` 3 N a 1 ^ ! !& ! ` 1^ 1 !( 7 m !( 1 = !& !C@ m 1! ( " !& !( N a 1 !& !( 1 = m !C@
h7 h7
9.
m !( " !& ! ( "
B B
B
B
+
:1 !< ;
m ! (E $ 5 m ! m ! ( m ( 7 ! " ! N a !& ! 11. ` 3 7X N a 1 ^ ! !& ! ` 1^ $ 1 m m !95 7 ( m B ( 7 ( ! !& & ! _ 5 7 a N 1^ ` !& ! !
(E $ 5:7! !& ! ` m !V5 1 = !& !C@ !& ! 5 1 = m !C@
( 7 ( ! ! " ( 7! m ! (K $ 5 7! '%& N a ! !& ! 5 7 ! ` $ 1^ ' ( O'7X! ! % £ &- O/ O$*' L F G F 7 ( $ G J m !95n')( = !& !C@ Y
!& !95:' ( = m !C @ ( ' ( ')(:7 ! ! 7 ! K ( _ 5 X 7 ! $ m ! $ (87!A $ 5 ! N a 1 !& ! ` ^ #$ !95 #7 ( m !95 1 = !& !C@ m 1 ! 3 = m !C@ !& !95 1 N a 1 !& !9 5 1 & ( 7 (
! ! $ 5_7 m ! (E $ 5_7X! ¢ ! !& ! (:7X!A $ 5 ! N a 1 ` 43 $ ^ 3 N a m ! ! ( ! ` ( 7 ! ! 1 ^ !& ( 7 ! " m ! (K $ 5_7! ! !& ! ` $ 5 !A $ (:7! N a 1 ^ ( 7 ( 7 3 7 ¢ N a m ! ! ( ! ` ! !& ! 1^
12.
13.
7 m ! N a ! : 5 7 ! $ 1^
3
14.
16.
7 B
15.
B
;93 N
B
7 B
7 B
7 m ! (E $ 5_7X! m ! ( ! !& ! ! N a $ 5_7X! N a 1 $ 17. ` 1 ^ !& ! ` 3 ^ 1 !95 V7 ( #$ m !95 1 = !& !C@ m 1 £ 3 / m !C@ !& !95 a 1 N a !& !9 5 ! 1 & 1 =
m m !95_7 ( B ( 7 ( ! !& N a 9 ! _ 5 7 ` ! ! 1 ^ !&
18.
19.
¢ $ $$ 55_7 7! 4 3 1 £ 3 a 1N a
7 $ 5 7! m
m
3
20.
m
m = m
( X7 ! " 5 7 N a 1 $
m ! (K $ _ 5 7! m ! ( ! !& ! ! N a N a ` 1^ 1 ^ !& ! m ! & ( #$ ! (' ( 7 !& ! 1 ! !
m ! (E $ 5_7X! ! !& ! ` N a 1 ^ ! W #$ £ m ! W & fI 0 N a £ !& ! 1 & fI 0 m !95 V7 W !95 1 = !& !C@ 1 ! W
a a N
!& !95 1 & !C@ !& !95 1 1
( 7 (
! ! m ! (E 0 $ 5_7X! 5 m ! m ! ( ! ! N a !& ! ` 1 ^ !& ^ ( 7
` 7 B
#$
B
!
!
m ! ` 7 B
6.17.6. Series containing p Fp+1 ((ap (k)); (bp+1 (k)); ϕ(k)z)
4-4# $ & 7,5 7 (8 $ " m !" ! 1. $ % ! " N a 1 1
1A
=
m 9! 5 $ ( $ !95 $ ] ! $ 5_7 ` ^ 7 a N m ! #7 ! ! ` 3 7 1 ^ ;931
= g M !
7R@CB
+
2.
3.
m !95 $ "" m ! " $ ! % ! " N a 1 !95 ^ 1 m !95 $ "" m ! " $ ! % ! " N a 1 9! 5 ^ 1
:1 !< :
m $ ( $ $ ! $ 5_7 `
$ ( $ $ ! $ 5_7 `
6.
m 9! 5 $ ( $ ! a N a 1 ^ !95 $ $ 5_7 ` m m 5_7X! _ 5 7X!A m n 5 ! m m 5_7X! _ 5 7!A n 5 !
= g M ]!
7A@CB
7R@CB
m !" m !95 $ ( $ $ " !95 $ ]! $ 5_7 ` $ ! % $ ( m ! ! " N a 1 ^ 1 7 ¢ 3 a N a m ! #! 7 #( 7 ! ( Vm 7<5 = g M ]! m m m ` 1 ^ m !95 $ (E $ 5_7X! $ 5_7X! "" m ! " $ 5_7X! % ! " N a 1 !95 ! $ $ 5n ` ^ m ¢ 3 = g !
5.
!"
m 5_7X!
= g M ]!
!"
5_7X!
4.
= g M ]!
m m 3
$ "" m $ ! % 1 m 3
7A@B
m !95 $ (E $ _ 5 7X! $ 5_7X! "" m ! " ! $ 5_7X! % ! " N a 1 !95 $ $ n 5 ` ^ m m 5_7! m 7 ¢ 3 7
5_7!
7A@ B
7A@ B
7.
;93,
= g !
7A@ B
8.
9.
$ 5 7 ! "'& $ 5_7X! % = $ 5 7! 5 m @ 7_ m 5 7a
$ 5_7! "" &$ %
m !" !" N 1 N a
m 9! 5 $ (E $ 5_7X! a 1 ^ !&5 ! $ $ 5: ` m ! #7 m ! ! &- ! & a = g
( $ m ! a 1 N ! ! a 1 (:7X! m !(:7 & 2- 1 !(8! 7 ` 3 ¢ N m (:7X! ^
(
7A@ B
!
&
7B
6.17.7. Series containing p Fq ((ap (k)); (bq (k)); ϕ(k)z) and special functions 1.
$ 5_7! 3 " " 8 " G a / & $ % F " " J L ! m
2.
( $ & m ! ¢ Q a N ! a ! 1 1 (:7X! !(87 ( W #$ m ¤ ¢ 3 N a ! 7 ! : ( 1 (:7!
$ 5_7! 3 " " 8 " $&% F3 G a /
" " J L ! m
¢ Q a N 1 (:7!
3.
(:7X!
( $ & m ! !! a 1
m ! (:7 N a 1 !(:! 7
W #$
3 ¢ ¤
m !" m !95 $ $ 5_7!0 $ 5_7! " " $ ! % F $ 5_7 G ! " N !9! 5 $ ` ^ (:7! " " m !(:7 ¢ 3 N a 1 !(87 ! V( m (:7X!
;93 3
¤ 1
+
4.
:1 !< <
m !" m !95 $ $ 5_7X! $ 5_7! "" !9! 5 $ $ 5_7! % a 1 F $ 5 7 G ! " N ` ^ (:7! m !(:7 Q& -21 ¢ ¤ !7 N a 3 ! : ( m (:7X! 1 1
5.
$ 5_7! ""
m !" m !95 $ $ 5_7! !9! 5 $ (P5_7X! " 43& c - F $ 5 7 G ! " N ` ^ (:7X! m !(:7 d, Q& -21 ¢ ¤ ! N a m (:7X! 1 ^ !(:7# ` 3
6.
( L" 1 ¢ -21 CJ 7 (* 9! " m ! Y ! " c - "
F G
m !95 $ $ 5_7! !9! 5 $ ¢ ¢ Q N ^ ` (87! " ¢ 3 a N a m !(:7 V('6( 1 ! ( P5_7!# !(87 V ( m (:7! 1 1 & 7 B m !" m !95 $ $ 5_7X! $ 5_7! "" C7 (&! " 43& c - F $ 5_7 G ! " N !9! 5 $ `
^ (:7! " " ¢ 3 a N a m !(:7 # Q(:7 ! ¤ #( Q(:7X! !(:7 V( 1 m (:7X! 1 1
m !" m !95 $ $ 5 7! $ 5_7! "" C7% ( 9! " 3P& c - a F $ 5_7 G ! " N !9! 5 $ `
1 ^ (87! m !;(:7 # Q& -21 ¢ ¤ ! a N a 3 ! : ( 7 m (:7X! 1 1 1
7.
8.
;9395
9.
$ 5_7! "" . g5_7X! " j + - l F $ 5_7 3 ! " ,
5 m
10.
! ¢ G m " N m !95 $ !&! 5 $ $ 5_7X! ` !" ^ (:7X! m !(:7 5 ¢ ¤ a ! : ( 7 3 a N 1 1 (:7X! !
$ 5 7! " ( Q( 2! " F G m !" m !95 $ $ 5 7! Y j + - l - . ¢ 3 ¢ ! N !9! 5 $ ` " ^ (:7X! m !(:7V( (:7 " 5
g _ 5 X 7 ! ¢ ¤ # 65_7! ! a N a m (:7! 1 1 ^ !;(:7 V( ( (:7 ` 3
11.
( 7X! " $ 5_7X! " ( $ : g5_7X! " 3 & 5 5_7 m ! " m !95 $ $ 5_7X! Y j + - l . F ( $$ : !&! 5 $ ( 7G ! " N ` ^ (87! " , ! 7 ¢ 3 a N a m !(8 7 V!( (: 7(8 5 _7 1 1^ m (:7X!
` ¤
. $ 5_7! " O ( ( ! " F G j + - l V- ¢ 12. m ! m Y ! " N " ^ (87! 5
g 5 7 5_7 O ¢ m (:7X! 3 a 1 N
!
;93;
¢ Q
!95 $ $ 5_7!0 !9! 5 $ ` m ! (:7V( : ( 7 ( " ( 7 ` a 1 ^ !(87 V( ( ! :
&
7B
+
m !" $ 5_7X! "" 5_7! " 3P& ! " 13. Y j + l V- . F ¢ $ 5_7 G N (:7!
¢ 3 a 1 N a 1 5 ! m (:7X!
( ]! " ¢ -21 1. $&% Y a N 1 (:7! Q& -21 m (87!
m !95 $ $ 5_7! !9! 5 $ ` ^ m !(87 5 !(:7 ( !
&
7B
6.17.8. Series containing products of
:1 !< L
p Fq ((ap (k));
(bq (k)); ϕ(k)z)
!" ! " ( $ & m ! 9! 5 $ $ 5_7!0 !9! 5 $ ! N ^ ` ! a (:7X! 1 m !(:7 !(:7 a N a !(:7 !(8! 7 ` 3 ¢ (:7X! ^ & 7 B
;93 :
Chapter 7
The Connection Formulas 7.1. Elementary Functions 7.1.1. Trigonometric functions
. + & -21 C 7( '! " £ C 7( '! " &- -21 A&- -21 ¤ & $ % '! " £ & -21 C7 (* ' $ ! " 5_ C7<75n ! % £ a 1 a 1 ¤ ('! " '65_7X! " £ ¢ 4 £ ¢ $ 5_7! % £ a 1 ¤
1.
2.
3.
4.
5.
6.
7 $9% C(7'(*! '" ! "
£ (' ! $ " ! '% ! " £
£ ¢
£ &-
&- ¤
¤
( ' ! " '65_7! " £ $ ! %
¤
7.2. Special Functions 7.2.1. The psi function ψ(z) 1. 2. 3.
7 F G F G , 3 . + & 3 ¢
G
¤
G
¤
&B B B ] 7 a 1 -;&-21 a N a 5 7 7 ]] #5_ 7&B B B #! 5_7 L 1J ;93<
= '8>?7A@CB
#
< , ,
7.2.2. The incomplete gamma functions Γ(ν, z) and γ(ν, z) 1.
( 7! ' & -21 < )3 & C7( ! ' & 3h -21 - ¢ 3 3& - ¤ < & & a &-21 - & -21 ¢ 3 T3 43& -
7 7 F G / F G o / 7 3 '65_7 a 1 - F]3 3 G 1 7 < ( 7! ' " o F 3 G / / -;&-21 - 1L' 1 (*' 1 J L" J
2.
3.
4.
¤
¤ 3P& ¤
7.2.3. The parabolic cylinder function Dν (z)
2.
3.
4.
5.
' a 0 & £ -; J $ L £ 43 &- F G - 0& ¤ ( 7 ! F G ' $ L 43 / £ A F G a & ¤ 0 & ' -; &- J W 0& £ -; - F G ¤ W ' 3 " ' 8 ' F G ' G % $ L 0 & 4 3 A F V- -;&-21 -21 & " J ' & 1 3 8 W ' % F G F G ¤
1.
W "
&-21 0& ( 3 £ " J &$ % L -21
&- ) W " ( J L £
&-!&4 0& 3 * $&% $ Y L S - a S J
6.
;93 L
,
S J
$
L S - a 1 )
¤ ,
-21
&- ) , ¤ &4 ) , 3 S - a 1 ) ,
W " - &-2]' "1 0& ( ' % J 9$ % L " 1 J L ' ']" ' % - &!- &4 0& & L ' * J
$ Y S J
7.
8.
$ -21
&- ) 3 , S J W " ( J L a 1
$&% &- ) 3 L S - a &4 ) , 3
¤ L S - a 1 ) , ,
¤ S - a 1 ) ,
7.2.4. The Bessel functions Jν (z), Hν(1) (z), Hν(2) (z), Iν (z) and Kν (z) 1.
2.
3.
4.
5.
6.
( 1;- ! ( ( ' #7 ( (*' & 1;- #7( ! (
N & 7 (*' 5_7# 7( (*' -21 & = '*>h7 ! = @ VB B VB #Bk !C@MB ( ( 1 ! 3 ; F 3 G N & ( ' ( (*'65_7 & -; & #7( ( ; 1 ¢ '*>?7R@CB & 2 1 ! 3 3 &-21 F 3 G
N & 7 (*' V7 ( (*'65_7 a 1 & = ')( $ 'g5 $ ! % a 1 & 43 ¢ * 3 ¢ $&% ')( $ ! % £ & - FO$ G ¤ ¢ 'g5 $ ! % £ 'g5 $ ¤ ) ' ( ! G FO63 & $ % $ % & 3 & * -;&-21
S . & -;&-21 F G + S -21 . + -21 3 ¢ a $ L &-21
J S ' %
W U a [ ¢ ¤ a Y & % B B B ( % 3 0 & Z Z Z
-21 [, 1 a U a ZZZa 1
1 £ & £ -;&-21 & 1 ' $ L& &-21
-21 & &- -21 & J 3?3 ¢ 1 - 0& -; a 1 0& ¤
a & ; F G N & 3 ¢ &-21 F G &-21
;93 P
#
< , 5
+ S -21 . + S -21 . ¢ 5 7 7. ;- &-21 F G 1 ;- & 3 J $ L S ' %
a 1 W W a [ ¢ a Y & % B B B ( % 4 3 0& ¤ Z Z Z
2 1 U a a [ 1 ZZZ 1
a
X@CB 8. a 1 & -; &-21 & a 1 A& a 1 3A& = ' % ! 9. '#" - &-21 - &-21 £ A& - &-21 43 £ A& ¤ ('%& ' & 10. &-21 " 5 ! 3 8 ' O'65_7 Y $ L F G D 43 ¢ F G &- -21 -21 J 'g5 7 3 43 ¢ F G -; a 1 1 - H '65_7 3 9D 3 ¢ F G &- -21 -21 O'65 7
¢ ¤ 43 F G -;&-21 1 - H '%& ' & 11. 1 -; " 5 ! 3 8 '65_7 ' Y $,L F G D F G a -; 1
-21 J '65_7 ¢ 3 &- F G &- -21 1 - H 'g5 7 9DC3 F G -; a 1 -21 O'65_7
¤ ¢ 43 &- F G &- -21 1 - H . + 1 / 12. &-21 7 . & -21 ¢ '65 $ (:7X! % " = '*>?7R@CB ) ' ( 8 ( 7 ! 9 $ % $ % 3 2 1 + 9-;
! * 13.
'65 $ (:7X! % 7 . . + 0/ * -21 # - + &-; &-21 $9% ')( $ (87! % ! " &-21
;!5N
= '*>?7R@CB
14.
15.
16.
( 1;- ! & (' V7 ( (*' & 1;- #7 ( ! ( ' 2- 1 &
N & 7 (*' 5_7#7 ( * ( 1 G N & ( ' ( ( ! 'g5 7 & #7( G &-21 N 7 (*1;' - #7 ( ( ! 'g5 7 a &
& 1 & 43 ¢ a 1 & a 1 43& =
a & ; F 3 G N ¢ - &-21 F]3 G & -21
= '*>?7R@CB
;- & 3 ; F]3 ¢ 3 ; &-21 F]3
= '*>?7R@CB
# a 1 & <3 # ;- &-21
17.
18.
19.
20.
21.
22.
23.
@CB
3 ']' " % ! - &-21 - &-21 £ & - &-21 3 £ & ¤ ( 1- ; F G N & ( ' #7 ( ! (*' & a & #7 ( ; 1 ¢ '*>?7R@CB & 2 1 ! &-21 F G
N & 7(*' 5 7 #7 ( (*' -21 & = ( ; 1 3 ; F G N & (' V ( (*! '65_7 0& -; & #7 ( ; 1 ¢ '*>?7R@CB & 2 1 ! 3 &-21 F G
N & 7 (*' #7( ( 'g5 7 a 1 & = a 1 & " O 43 £ & -; - - &-21 £ & ¤ &-21 & . S ' % -;&-21 IF G + S -21
% B BB ( % -21 & ¤ , [ U a ZZZa 1 £ & 63 £ -; & 1 43 ¢ ' $L ¤ & 0 &
&-21
< 1 2 1 J ¤ £ -; F G 1 ' $L 0 & 0 & 2- 1
&- -21
J ;!5)
24.
25.
#
< , ;
( 7 1;- ! ( £ a / / ;- N & (' V( (* '65_7 ; ' ( ' * ( ' 5n' Y & F 3 3 G ' "' ¢ 1;- #7 ( ! ( 3 3 N 7 * ( ' 5 7 V( (*'65_7 9l & 7 7 7,5 7 ( Y D & F ¡ ¢ 3 G & F ¡\ IG H ¤
-; / / ']" 43 ¢ 43 ¢ 3 3 9l N & 7 (*1;- '
7( '
( ( ! 6 -; N & ( ' ( 1; - (* ' 5_7 #7 ( ! ( F ¡ 7 3 QG #7( ( 'g5 7 & # 7 ( ( 1;- ! 5 N 7 (* ' # 7 (
(*'65_7
& 7 7 7,5 7 ( Y D & F \¡ \¢ 3 QG & F ¡\ IG H ¤
7.2.5. The Struve functions Hν (z) and Lν (z)
¤ ( 7 ! ' & 2- 1 1 L "
& -21 J
G 1. H & <3 F 1 J L ']")" '#" ¤ 7 L J 2. H a 1 0& a 1 0& ' % 43 F G F 3 G 43 ¢ a 0& ¤ & 3. H
-;&-21 1
( ( ; 1 ! ; F G N & ( ' V7 ( (*' H 0& 4. H a 0& V7 ( ( 1 ¢ & 2 1 3 - &-21 F G N & 7( ' 5_7 # 7 ! ( ( ' H -21 0&
'&(']" ( 1;- ( C7 ( ( '! " L & 2 1 G N & ( $ ( $ 5: ' V! 7 ( J ( $ 5n'65 1 L F]3 J
4 3 ¢ H & ;-
;!5,
( ' = '*>?7R@CB
( ( ; 1 ! F 3 G N & ( ' V ( (*'65_7 H 0& #7( ( ; 1 & 2 1 ! F3 G
N & 7 (*' V7 ( (*'65_7 H a 1 0& ( 1;- ( (*'65_7! " '8>?7A@CB ! 5 $ (*'65 L N & ( $ ]')( $ ( ( 'g5 7 = J
¤ 7 & -21 ¢ 1 L "
& -21 J , G
6. L & L & -; 43 1 L '#)" " F J
']" 7 L F G F G ¤ J ' % & 0 & 3 4 3 7. L a 1
-;&-21
¤ 8. L -;&-21 & a 1 0& ( ; 1 ; F 3 G N & (' V7 ( ! (*' L 0& 9. L a & V7 ( ; 1 ¢ & 2 1 &-21 F 3 G N & 7 (*' 5_7 #7 ( ! ( ' L -21 0&
!& '#" ( 1;- C7 ( (*'! " L & 2 1 G N & ( $ ( $ 5n ' #7! ( (*' 3 J ( $ 5:'g5 1 L F J
= '*>?7R@CB ( ; 1 10. L 3 ; F]3 G N & (' ( (*! '65_7 L 0& -; 0& #7 ( ; 1 ¢ & 2 1 3 ; &-21 F]3 G N & 7 (*' #7( ( ! 'g5 7 L a 1 &
"'%& ( 1;- (*'65_7! " L & 2 1 '8>?7A@CB ! J 5 $ (*'65 L N & ( $ ]')( $ ( ( 'g5 7 = J
5. H ; - 0& 4 3 3 ¢ 3 &-21 "'%& L & -21 J
7.2.6. The Anger Jν (z) and Weber Eν (z) functions
¤ - 0 & J 43& ¤ J 0& 0&
1. J 2.
;!53
#
< , :
#7 ( ( &-21 N 7( 1' - 5_7 # 7 ! (*')( J &
& 7( (
N (*' 5n&< - V 7 ! (*')( J a &
& 1 ( F G a 1 N ( ( 1;5n- ' V! 7 ( (* ')(
& $ $
= '*> @CB ( ( ; 1 ! 43 F]3 G N & (' V ( (*'65_7 J & 4. J -; 0& #7 ( ( 1 ¢ & 2 1 ! 3 3 &-21 F3 G
N & 7(*' #7 ( (*'65_7 J a 1 & ! ( ( a & -21 1- ! ¢ ¢ 1 G 3 3
N & $ ]')( $ ( ( 'g5 7 T3 F 3 = '*>?7R@CB 3 E 43& ¤ 5. E 0& - . 7 # ' " 7 +
& 2 1 ')(:7! % % G G G 3 H 0& ¤ 6. E 0& F F 3 F]3 . 7 '#" 7 +
& 2 1 ¢ O ) ' : ( X 7 ! G G G % % F F 3 F3 43 7. E & -; 3 43 ¢ H-; & ¤ ¢ ¢ ¤ 8. E a 1 0& 43 ¢ J -;&-21 & 3 ¢ J a 1 43& ¤ 9. E a -; 1 0& 43 J&-21 & 3 J -; a 1 43& #7( ( ; 1 ¢ & 2 1 ! 10. E a & &-21 F G
N & 7 (*' 5_7# 7(*')( E 0& 7 ( ( & £ 3 &- F G &- N & (*' 5n #7! ( ' ( E a 1 0& 7 & - ¢ ¢ a ¢ 3 )3 43 ( ( Y F]3 G a 1 N ( $ ( $ 1;5n- ' #! 7(*')( = '8> X@CB
& - ¢ F G 3. J a 0& & -21 3 - £ &- F G & ! & - ¢ ¢ 43 3 )3
;!5 5
( ( 1 ! G N & ( ' ( (*'65_7 E & 11. #7( ( 1 ; & 2 1 ! G
N & 7 (*' V7 ( (*'65_7 E a 1 0& a 43 ¢ a )3 ¢ F G 1 ( ( Y N ( $ ]')( $ 1( - ! (*'65_7 = '*>?7R@CB
& ( 7X! ' & -21 ¢ O')( $ 5 a G F G 1 12. J a 1 0& 3 F 7 ' -21 $ (:7 S 3 J $ L F G -21 J L F G S F G S - -21 Y &S 1 0& 43 ¢ a -21
a -; &-21 & &- a 1 0& ¢ ¢ S 3 3 1 - S 0& 43 ¢ -;&-21 & <3 &- a 1 0& a 1 & <3 3 -;&-21 0& & ¢ a 1 0& ¤ 43 -;&-21 & 0 & ( 7! ' & -21 ')(* $ 5_7 a G 13. J F F G 1 1 -; 0& 7 ' -21 $ (:7 S 3 J $ L F G -21 J L F G S F G S - -21 Y &S 1 & 43 ¢ a 0& 43 ¢ -21
&- -2 1 0& 1 -; ¢ ¢ ¢ S 43 & a 0& 3 43 ¢ ¢ S & 3 ¤ 1 -; 0& 3 41 3 - &-21 & 0 E1 0 & -; 1 -; & 3 &- -21 & -2 1 & 0 & E ; - & 3 ; F]3 3 ¢ 3 &-21 F3 7 & -21 3 ¢
7.2.7. The Airy functions Ai (z) and Bi (z) 1.
2.
7 " D 1 & F &4 G 3 1 & F &4 G9H @ B " D 1 & F &4 G 1 & F &4 GIH = 7 " D £ -21 & F &4 G 3 £ -21 & F &4 G ¢ ¢ / 1 & F &4 G 3 3 / 1 & F &4 G9H £ £ " D -21 & F &4 G -21 & F &4 G ¢ C@ B 3 / 1 & F &4 G ¢ / 1 & F &4 G9H = ;!5;
#
< , L
7.2.8. The Kelvin functions berν (z), beiν (z), kerν (z) and keiν (z) 1. 2. 3. 4. 5. 6. 7. 8. 9.
1 0& * F 3 G G F & * 1
& F
* -21
-21 & * F 3 " O #- A F G ¤ 0 &
1
3-" - A F G ¤ & 1
" O - A F 3 G ¤ &
-21
3-" V- A F 3 G ¤ & -21
F3 G a & #$ ( 1;- - &- ( ! Y N ( a 1;- 1 1;- E- #7( 1 a 1 ' ' Y D & <3 0& H
¤
¤
G ¤ G ¤
#$ - &- #7( ( 7 ; 1 ! & 2 1 3 F3 G N 1- #7 ( 1 a #7<5 1;- E- #7 1 ( a 1
'65_ 7X !
'65_ 7!
Y D -21 0& -21 0& H ')(:7 5n')(:7! F]3 G &-
#$ # 7 ( - (
< & ! Y N 1;- V7 ( 1 a V7<5 # 7( a &1 - E- &
' '
') (* Y D 0& 0& H 3 5_7X!9 5n')(:7X! ; F]3 G & -!&
;!5:
Y
N
#$ &- #7( - - ! ( 7( &- #7,5 & a # 7( a & 1 - E- &
'65_ 7!
'6 5_ 7X!
Y
D
10.
-21 & <3
a 0& ; ]F 3 G ( Y N ( 1; - a - &- ;1 - 1 1;- E' ' Y D 0&
n 5 7X! ' ' 5
Y
11.
Y
-;
3 4 3
¤ -21 0& H
#$ ! ( 1 #7( a 1
0& H #$ 7,( (
9 1 P & ! F3 G &-21 N 19- 7( 1 a 725 1I- - 71 ( a 1
' 5n 7X!
' 5n7X!
Y D & 3 H -21 & ')(:7 -21 ¢ '65 !& '65 (87! F3 G &-
#$ # 7 (
- &<- - ! ( 1;- &- #7 ( 1 a #7,5 1; - E- 1 V7 ( a & '
Y '
D & <3 0& H ')(* 3 '65 ! '65 ! (:7 - £ F]3 G &-!& #$ &- #7( - - ! ( N 7( #7,5 a #7( a 1 & &- &- E- &
'65_ 7X!
'6 5_ 7X!
¤ Y D
-21 0& -21 0& H & 3 ; F G $# ( 1;- - &- ( ! Y N ( 1;- ( 1;- 1;- a #7<5 1 - 1 ' ' Y D
& < 3 0& H
#$ #7( (
; 1 & ! F G &-21 N 1;- #7( 1;- #7( 1;- a #7,1 5 - 1
;! 5<
N
#
< , L
'65_ 7X! 1 & 0 a 0& H 1
')( (:7! 3 ; F G & #$ # 7 ( - (
< & ! Y N 1;- #7( 1;- #7( V 7<5 - &1 - a & ' ' ')(* Y D 0& 0& H (:7X!9 (*'65_ 7X! 3 F #$ # 7 ( ( Y N 7 ( &- V 7 ( - V 7<5 - ! 1 a &- - &- & &- '6 5_ 7X!
Y '65_ 7!
D a 1 & 3 a 3 F G 0 & 12. -; #$ ( 1;- - &- ( ! Y N ( 1;- ( 1;- 1;- a #7<5 1 - 1 ' ' Y D 0& 0& H
#7 ( ( C7 ( ! '
; 1 < & ! 3 (*' F G &-21 N 1;- #7 ( 1;- #7 ( 1- a #7,1 5 - '65_ 7!
'65_ 7X !
Y D a 1 & <3 a 1 0& H ')(:7 ¢ ')( !& ' ( (:7X! 3 ; F G &-
#$ # 7 (
- &<- - ! ( Y N 1;- #7( 1;- #7( V 7<5 - &1 - a & '
Y
'
& <3 0& H D ')(* 3 ')( ! ')( ! (:7 £ 3 F G &!- & #$ #7( - - ( & Y N 7 ( V7 ( V7<5 ! 1 a &- - &- & &- '6 5_ 7X!
Y '65_ 7X!
a 0& a D 1 ¢ £ 13. a 1 & 3 O '65 / 6 & -;&-2 ( 1 7 ! V" - A I " £ Y F $ ! % G - &-21 / &-
Y D
'65_ 7X!
a ')(:7
G &-!&
1 0& H ¤
#$
1
;!5L
¤ 1 0& H
3 14.
F
'65
. " "'& + & -21 ( 7 ! I $ _ 5 7X! % G
- / £ &- -21 =
£ & -;&-21 - A " a 1 & 3 ¢ a 1 / 6 " O'g5 ( 7! I Y F $ ! % G - &-21 / & -
' " & . " O'65 + & -21 ( 7X! $ 5_ I 7! % - / F G &- -21
@CB
£ £ =
@CB
7.2.9. The Legendre polynomials Pn (z) 1. 2. 3. 4. 5. 6. 7. 8. 9.
j & 4 3 ¢ j 3P& ¤ 1 & ¤ j + l A . 0 & ¤ 7 . 5 F (8 G j + lC- &-21 F 7(8 G ¤ 3 £ 3 ¢ j + -;&-21 lC-;&-21 . F G ¤ : ( 7 . 7,5n ¤ ¢ j & 0 3 j + -21 lC- &-21 7(* , ) j + lC- &-21 . F 3 ¢ G ¤ . 7,5 ¤ ¢ j a 1 & ;0 3 j + 1 lC- &!- &4 ) 7(8 , a 1 j + lC- &-!&4 . F 3 ¢ G ¤
7.2.10. The Chebyshev polynomials Tn (z) and Un (z) 1. 2. 3. 4.
0& 3 ¢ 3P& ¤
7 3 ¢
) * 7,5n
) * , 0& 3h &-21
(8 ¤ ,
¤
0&
;!5P
= '8>?7A@CB
+ 5.
6. 7. 8. 9. 10.
11.
12.
13. 14. 15.
16.
17.
18. 19. 20. 21. 22.
' c ' % 1 J L' 0&
#
7 ¤ c & . j + -21 lC-21 0& ¤
£ ¢ ¤ 3 3 ¢ F ¢ h 3 G ¤
£ 0& 3 ¢!¤ 43 ¢ a 1 7 7 ¢ £ * 3h 1 , ) ' (:7 ¤ % '65 1 L ' 1 - ) , J
' % j + -21 lC- . F 3 ¢ G ¤ 1 J L' 7<5 7(* £ a 0& a a
1
1 *
1 43 ¢ F ¢ 3h G ¤
' % a 1 -21 - (:7
, '65 & L '
) J
' % j + -21 lM1 . £ 3 ¢ ¤ 1 J L ' ' % a 1 j + -21 lC- &-21 . F 3 ¢ G ¤ 1 J 7L ' & 0& ¢ ¤
¢ 3 ¢ 3h
0& 43 ¢ 43&&-2 ¤ 1 & 1 & ¤ '65_7X! % j + 1 lM1 . & ¤ & J L' ; ;9N
< , N
7 7 ¢ 3 1 , ¤
) * 3 h
7 ( 7 (8 *
¤
¤
= '8>?7A@CB
( 7X! ' a F ¢ 3h G ¤ & 7 8 (
1 3 ' % O'65_7X! a 1 0 3 ¢ -21 -21 - ¢ 3 a 'g5 1 L '%&
1 ) * J
' % j + 1 lC-21 . £ 3 ¢ ¤ 1 J L' O' '% ! ! % £ & j + 1 lC- &-21 . F 3 ¢ G ¤ £ £ ¢ ¤
a 1 & 3 ¢ 43 ¢ 3h G ¤ 7 (8 a 1 F 3 # '65 7! % ¢ a '65 ! '65 & L' 0 3 -21 -! &4a 1 - ) * J
' 6 ' _ 5 X 7 ! % % '65_7! % £ & a 1 j + 1 lC- &- . F 3 ¢ G ¤ 7 ¢ & 7 8 ( 3 a 1 0& ¤ ¢ 0& a F ¢ 3h G a 0& F 3 G
1
1
&-21 3 ¢ ¢
23.
24.
25.
26. 27. 28.
29.
30. 31. 32.
2. 3. 4.
7 , ¤
¢ 3 7 ¤ ,
3h
= '8>?7A@ B
¢ 0& 3 43& ¤ 0& 3 ¢ £ -21 0
¤
a 1 & 3 ¢ £ a 1 1 ¤
¤4¤¤ ]S 0
1
[<\]W " U M (87! ']")" U " W M (:7! " U ")" W U £ ¤¤4¤ ' ( $ ! % $ ( $ ! % ¤¤4¤ U XW4 X[<\#U " [<\#U M (:7X! " [<\]W ")" [<\#U 7 Y ( " $ (( "" ( $ ( " ! % $ ( " ! % X[<\#U S ¤
7.2.11. The Hermite polynomials Hn (z) 1.
; ;
+
a 1 1
Y U4
5.
#
¤4¤¤ S £ ¢ ¤¤4¤ S [<\]W " U M1 (:7X! '#")" -2U 1 U ¤¤4¤ ')( $ ! % XW4 X[<\#U4 ( " " [<\#U M ( " (87! " [<\]W ")" [<\#U Y $ ( " ( $ ( " !%
< , !,
( 7X! " U ")" W ¤4¤¤ " W M : $ ( $ !% 7 $ ( " 5_7X! % X[<\#U a 1 0 S ¤
7.2.12. The Laguerre polynomials Lλ n (z) 1.
2.
3.
4.
5.
6.
( 7! ' ¤ ' -21 0& ' % ' 0/ ( 7! '
0 & & ' % a 1 / ¤ 1 ')( ! % ' % 3& S S&- S & - S & ( ]! ' ¤ ' % -; & ( 7! ' ' % &-21 Q3 ¤ 1A-; & ( 7! ' ' % ¢) & ¤ -;&-21 &
7.
8.
9.
10.
11.
=7
'I@CB
" 3 ¢ $9% ¤
( ]! ' ' % " A a 1 F ,G ¤ - &-21 & - &-21 & - &-21 3P& ' % ! a 1 D V-; &-21 F G 3 # a 1 F ,GIH ¤ ¤¤4¤ S c 1 X[<\]W U U U W ¤4¤¤ ¤¤4¤ c - c U -21 c U - c W -21
1 U W [<\#U &< [ ] \ W Y Xc [<\]W - Xc [<[<\#\#U U -21 ]S Xc [<[<\#\#U U 0SQ ¤ -21 4 ¤ ¤ ¤ c [<\]W " U C7 (8 ! '#")" U " W C7 (8 ! " U ")" W 1 S U d ¢ ¤4¤¤ ) ' ( ! $ ( $ ! % ¤¤4¤ $ % U XW X[<\#U " [<\#U C7(* ! " [<\#U 7 Y ( " $ ( " ( ($ ( " " ! % ( 5_7X! " [<\#U cX[<\#U S ¤ ; ; ,
7.2.13. The Gegenbauer polynomials Cnλ (z) 1. 2.
3.
c & 43 ¢ c 4 3& ¤ 3 £ -; 9! ' 3 ¢ 1 - c -; F G ¤ : ( 7 5 1 P ' L J
& ! ' . ¤ 5 1 j + c -21 l c -21 & P
L'
(9! £ 5:' '! ¢ j + c -21 lC- c - . F 7<5(8 G ¤ 4. ' & ! _ 5 7 (8 ' j + c -21 lC- c - . F 7<5 G ¤ G F 5. P5 1 L' J
£ £ 3 ¢ 3 ¢ -; j + c -21RlC- c -; . F # # 3 ¢ 3 G ¤ 6.
¤ 3 £ 3 ¢ j + - c -;9lC- c -; . F G 7. : ( 7 ( & ! ' . ¢ £ ¤ 43 ¢ j +
3 O 8. c & c 2 1 l 2 1
1L' J
( 9 ! ' . 7<5 ¤ ¢ 3 j + -21 lC- c - ) 7 (8 , 9. 1L' J
' % . O'! % d2 £ & j + c -21 lC- c - F 3 ¢ G ¤ 10. £ 43 ¢ ( &! '%& j + 1 l c -21 . ¢ 3 £
¤ 11. c a &
1 & J L' #( &! '%& . 7<5 ¤ ¢ ; 3 j + 1 lC- c - &-21 ) 7 (8 , 12. &L ' J
O'Q' 5_% 7! % d2 a £ & 4 a 1 j + c -21 lC- c - &-21 . F 3 ¢ G ¤ 13. 1 ¤ S - S S -; & 14. - -; &
S - S -; & = >?7A@CB 15. - a -; &
71
S -21 ' 0& ' >?7A@CB =8 16. c 0& c j & ¤ 17. 1 & J
; ;93
+ 18. 19. 20.
21. 22.
#
7 -21 & ' (87 j & -21 0& 3 j 0& 1 &
¤ 0 & 1 L ' 1 ;- & 3 £ J ' % 0 3 &L ' -;&-21 0& £ &-21 '6J 5_7X! % ¢ -; 0& 3 £ 0 3 ¢ j 1 -; &
24.
1 a -; &
¢ F G ¤ ( 7 :
3h& a 1 3 ¢ ¢ & a 1 ¤ F (:7 G
1 J ' L % ' ¢ h 3
¤
26.
1 J ' L % '
1 '65 J
1 ' &
'65 J ' % 1 L' ¢
27.
& L ' -21 - & £ &-21 OJ Q ' 5_7! % ¢ h 3 & a 1 ¢ & a 1 ¤
28.
29.
30.
31.
32.
= '8> X@CB
23.
25.
< , 3
; ¢ 3h ¤ 7 1L' ¢ h 3 F 7(8 G ¤ 7,5 ¤ 3h
) 7(8 , %
% 1 ( L ' S a ¢ 3h &- S &- SEa 1 & ¤ J * ;- 1 0& ' ( ' %
S 1 L( J
( 1 ( L '%& % S S a ¢ SEa ¤ -;a 1 0& ' % J ( 1 (*' L ( & 3h
&- &SE- a 1 1 0&
1 J
7<5 ¤ 1 L ' J 1 - &
43& 1 -; ) ,
1 J L' '65 1 L'%& 7 -21 a - & 3 ' J % '6 5_7X! ; ¢ 3h F 7 (8 G ¤
1
& L '%& !- &4a - & £ J O '65: O! % ¢ 3h& a 3 ¢ & a R ¤
1 ; ;!5
= '8>?7A@ B
33.
1 -; & F ,G 1 -; F ¢ 3 G ¤
34.
a -;&a -21 0& 3 F G 1 -;&-21 F ¢ 3 G ¤
1
35.
&L ' £ ¢ a
J
7X! % @ 3 1 a 1 ) -;&-21 0&
&-21 = '65_
36.
37.
c ¤4¤¤ ]S 1 U
[<\]W '#")" U " U ")" W £ d2 4¤ ¤¤ ) ' ( ! ( $!% $ % $ U XW X[<\#U4 ( " [<" \]W ")" [<\#U ( ! ")" [<\#U Y $ ( " ( $ ( " ! % C7 (Q (*'! " [<\#U
2. 3. 4.
5.
[<\#U c [
$ % $ $ % W [<\#U ( " " [<\#U C7 8 ( ( " ! " [<\]W ")" [<\#U 7 [<\#U ¤ $ ( " ( $ ( " !% C7(%6(*'! " [<\#U c a [<&\#- U S 0
4 ¤ ¤ ¤ ¢ ¤ 4 ¤ ¤ c a 1
S 3 d2 1 S -21
1 [<\]W " U C7 (8 ! '#")" U " W C7 (8 ! " U ")" W U Y ¤¤4¤ ')( $ ! % $ ( $ ! % ¤4¤¤ U4 XW4 X[<\#U4 ( " " [<\#U C7 (8 ( " ! " [<\]W )" " [<\#U 7 X[<\#U Y $ ( " ( $ ( " !% C7% ( 6(*'! " [<\#U c Xa [<&\#- U a 0 SQ ¤
1
7.2.14. The Jacobi polynomials Pn(ρ, σ) (z) 1.
¤4¤¤
c 4¤ ¤¤ ]SQ 3 ¢ d2
1 X [<\]W " U C7 (8 ! '#")" U " W C7 (8 ! " U )" " W U ¤¤4¤ 4¤ ¤¤ Y ')( ! ( !
U Y
38.
_ 5 7 (:7 , 3 ¢ ¤
. . j + l & 3 ¢ j + ]l 43& ¤ . 5 F 7 (8 G j + - - V- &-21Rl F (:7 G ¤ ( ¤ 3 ¢ F 7<5 G j + - - - &-21Rl . F 5_ 7 G 5_7! ' . j + l & 5_7! ' a 1 0& ¤ 3 ¢ (87 - -; F G ¤ (:7 , ) ; ; ;
+ 6.
( 7! ' ] C7 (8 ]! g 5J J L' 3 £ -;&-21 g5nJ '6& 5_ 7X! '%& .
j + 1 l 0 &
#
7 (8 a a 1 * , ¤ 1 L '%& 1 )
< , 5
&L '
M _ 5 7! '%& (:7 ¤ 5_7 ,
7. & 2 1 M E(:7!
a 1 ) * 7(* ¤ 1 . ¢ J 7 L ' O! ' a 1
65_ , 8. j + -21 l 0& 4 3
) * 5_7 (:7 1L' g5nJ '6
5_7X! ' F G - - ) * 5_7 , ¤ 9. . O'! % 5_7X! ' 5_7 a ' % 5 7! ' F G 1 * 5_7 , ¤ 10. j + lC- - &-21 0& ) 7X! ' 7 (8 ¤ O' '% ! % 5 5_ - - 7 ! 7 * ( G 11. F ' ,
) * . O' 5:7! % 5n7X! ' 5:7 a 1 a ' % 5n7! '%& F G a 1
: 5 7 , ¤ 12. j + lC- - &-!&4 0&
1 ) * 7X! % 5 7! ' 7 (8 a 3 ' '6% 5_ 5_7X ! ' & F G 1 - a - &-21 * 7(* , ¤ 13. ) 1 _ 5 X 7 ! _ 5 7 (8 ' . a 1 F 7<5 G ¤ _ 5 X 7 ! G 14. j + lC- - &-21 0& F ' 7 8 ( * ( 43 ¢ F G - -; ¤ 15. 7 * ( , ) _ 5 a . . £ !- &A 7X! ' ¢ 3h& -;&- F E5(:7G ¤ 16. j + lC-;&- + 1 & 5 7 L ' J
7! 5 7 S . ' % % '65 5 5_ &- S j S + l &- S . 0& ¤ _ 5 X 7 ! G F 17. j + l -; 0& : ( X 7 ! % '65 5_7X! S -; . 0& S . >?7A@CB + C l ' 5 ! j % 18. j + lC- - -; & = S -21 n 5 ' ! % S .S ' % 5n'65_7X! ( F 5_7G j + lC- S . & ¤ 19. j + l & SEa . 5_7X! ' (:7 ¤ ' % F G 20. j + -;9l & A. j 0& ¤ 21. j + l 0&
; ;9:
22. 23. 24.
25.
26.
27. 28. 29.
30. 31. 32. 33.
34.
35.
36.
37.
7 . j + Cl -21 0& j & -21 & j 7 . j + lM1 0& 7 (8 j 0& 3 j . j + lM1 0& F 5_7 G 1 j a 5_7 . j + lC-21 0& j * , )
& 0 a 1 0 & ¤ _ 5 7 ¤ 1) * , ¤
&L ' . j + 1 lM1 0 & '6J 5_7X! % 0& ¤ 1L' . j + -21 lC-21 0& J ' % & ¤ . . j + -21 lM1 0& 43 ¢ j + 1 lC-21 3P& ¤ 7(8 ¤ 43 ¢ J 1' L % '
) * , 5 7 J '1 L % ' F 5_ 7 G 1 a 1 * , ¤ ) 5_7 5 . j + lC-;&-21 & F G j 5_7 , ¤ ) _ 5 7 ( . j + lC- &-21 0& 43 ¢ F G j F 5_7G ¤ 5_7 . j + lC- &-21 0& F G j * 5_7 , ¤ ) 5_7 a . j + lC- &!- &4 0& F G 1 j a 1 * 5_7 , ¤ ) '! % 5_7 . j + 1 lC- &-21 0& ' % ! F G ) * 5_7 , ¤ ( (:7X! '%& (:7 . £ j + 1 lC- &!- &4 0& 43 -; C7(8 ! j a 1 ) * 5_7 , ¤ '65_7X! % 5 7 a . j + 1 lC- &- 0& ' % '65_7X! % F G 1 a 1 ) * 5_7 , ¤ ; ; <
= '8>?7A@CB
+ 38.
39.
40. 41.
42.
43. 44. 45. 46. 47. 48.
49.
43 ¢ '6J & 5_L 7X' ! % F 5 7 G
#
( F 5_7 1 L ' 5_7 . j + -21 lC- & 43 ¢ J ' % F G
< , !;
G ¤ E( ¤ 5 7G F _
7 J '1 L% ' F 5_ G * 5_7 , ¤ ) 5_7 (:7 . j + -21 lC- &-21 0& 43 ¢ F G j ) * 5_7 , ¤ 5_7 (87 & L' . j + -21 lC- &-21 & 43 ¢ ' % J '65_7X! F G ) * 5 7 , ¤ 7 a J '1 L% ' F 5_ G 1 a 1 * 5 7 , ¤ ) 7 . j + -;&-21RlC-;&-21 & F3 G 3 ¢ F (87 G ¤ 7 . j + -;&-21 lC-;&-21 & F G 3 ¢ j F (:7 G ¤ 7 . j + - &-21RlC- &-21 & £ - ¢ 3h a 1 a 1 F 7 (8 G ¤ 7 . j + - a &- lC- &- & 3 £ - &-21 ; ¢ 3h O a 1 a 1 F 7 (8 G ¤ 1 ¢ & j + lC- . & j + - -21Rl a 1 . & ¢ 3h& j + - l . 0& j + a 1RlC- -21 . 0& ] ( ! ' ' ! 5_7! ' ¤ % . j + l 0 1 ¤4¤¤ S '#")" U W " U ")" W [<\#U " [<\]W ")" [<\#U [<\]W U U ¤4¤¤ J ') ( L $ ! % J $ L ( $ ! % ¤¤4¤ J $ ( " L ( $ ( " ! % U XW X[<\#U4 Y ¢ X[<\#U j + [< a \#&U - [<\#U l a &- [<\#U . S ¤ &-
7.2.15. The polynomials of the imaginary argument 1. 2.
j A& F 3 A& 43
G ¢
¢ F ¢
-; F 7<5 G
¤
G
; ;9L
= '8>?7A@CB
3. 4. 5. 6. 7. 8.
4 3 ¢ A F ¢ G ¤ a A &
1
' ( 7X! a F ¢ G ¤ A & 7<5 1
43 ¢ a F ¢ G ¤ a A & 7<5 1
1 A& 43V -21 43 O ¤ ¢ £ a ¤
a 1 A& 3 ( 9 ! 1 A 1 43
' c A& 3 £ A 5n'! ' ¢
1 - c -; F 7<5 G ¤
7.2.16. The complete elliptic integral K (z)
7
1.
2.
7 G KF 7( K *
( *
3. K
/ £ 3 ¢ (87 K ) ,
5. K 6.
*
4. K
(
7. K
3 £/ £
8. K
)
" £/ £ 3 £
9. K
)
10. K
)
"
£"
3 £/ £
/ £ 3
F
7 ¤ G
" 7 ¤ F G F G " " 7 7 ¤ F G F G
" F 7 G ¤ 7 7,5 F G F 7 & F G 7<5 7 2 ;F G 7 5 F , 7 5 F , 5n , ; ;9P
G
= @ B
= @ B
= @ B
G F G ¤ G F G ¤ 7
F G ¤
+ 11.
12.
15.
16.
17.
18. 19.
20.
#
< , !<
5 7 ¤ ¢ £ £ ¢ " / K 3 , F G ) 5n 7 ¢ L ¢ £ £ 2 IF G ¤ / , K " 3 )
£ 3 / 5 7<5 7 ( G F F " 7 ( G G :F G K F F , ) " 7<5 ( 7 6( L : 7 J K 2 IF G 7,5 L ( 7 J 2 F G K 7,5 L J (* L 6(* 5 7 J 2 IF G K 5: 5_7 7 £ £ £ K P3 / 3 / & F G £ 3 ¢ & £ 3 / G KF / " 7,5 5 6(:7 7 F G ( ( ( K , ) " " 7 F 7 G F 7 G
13. K
14.
/
3 / £
21. K
g (
g (
6( 5
" £ £ 7 / /
7.2.17. The complete elliptic integral E (z) 1. E F
7 7 7 D 2 ] G
2
H ¤ ;9: N
£/
7O 7 G
= @ B
= @ B
= @ B
= @ B
= @ B
= @ B
F G
= @ B
F G
= @CB
7 7 1 F G F G ¤
2. E 3. E
)
4. E
)
5. E
)
6. E
7. E
8. E 9. E
)
7 7 3 £ / £ * D 2 IF G / £ 3 ¢ 2 IF GIH ¤ 6: ( 7 ¢ 7 £ £ £ ¤ " 3 / , D / F G F G F G F GIH ( 7 £ £ £ D F G F G F G F G9H ¤ " / 3 , ( D F 7 G F GIH ¤ £" / £ 3 , 7( " 7<5 7 G G G G! ¤ F F F F * " 7<5 7 ( 7 G G < G < G ¤ F F F F * ( " ¢ £ F 7 G F GIH ¤ / D * " 5 7 ( F G F G F G , ¤ F G F G :F G
7.2.18. The Legendre function Pνµ (z) 1.
2.
3.
( ) (* ' £ F * G ¢ 3
' ( ! " j -; & £ & - ¢ 3h
-
$J L a
- 1 L " l -21
J 5 Q(87! Y -21 % $ $ ( Q(:7X%! % £ & - ¢ 3 j a -;&- & ¤ -21
a " £ J & - L ' ¢ 3h ' $ L J 1 ( L' 1 5 L' 1- - L " J
J J
Y j a 0& 43 £ & - ¢ 3h
-
l -21
5 (:7! 7 G £ & - ¢ 3h
j - a a 0& ¤ Y -21 % $ $ ( Q(87%! % F 3 : 3 -21
¤ j & j & -;&-21
&-21
j
& &-21
Y
;9:1
Chapter 8
Representations of Hypergeometric Functions and of the Meijer G Function 8.1. The Hypergeometric Functions 8.1.1. The Gauss hypergeometric function 2 F1 (a, b; c; z) 1.
2.
3.
4.
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N 1 m 5:'g5 7 ! J
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5 1 5 1 N 1 '6 5 &
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28. N
6 ' 5 6 ' 5 (
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J m (:7 (:7 m (:7 (:7 ¤ Y N
1 m 5 ( 1 ! 1 a 3 m 5 ( 1 ! 1;-
a m V7 ( m ¢ - -Q1 @ B 7!C@MB 2 1 L = = h 3 & N & C 7 8 ( !
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( m (:7! ( (87! (:X7 ! ( m ( (87! = ( m ( !
& (* m L
m ! C7 * ( m!
= = @ B 7!C@MB
= = @ O !C@CB
5 1L 5_ 7X!
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88.
7
N &
87.
1
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85.
1 1 1 & #7V 7 & (
89.
!
$# & & &
m m m 5_7 #7#7 ( 7 !
m 1 1 ! (
& ( m 1 #7
#$ 1 1 1 1 #7 V7 ( 7
82.
£ 3 ¢ ¤
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! ] (* ( 7 m N & J (:7] ( m (87 ! ( (87 L N &
79.
:1N
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#$ 1 1 7X 90. N
& & 1 # 7 V 7 ( & 1 ! 1 1 #$ 1 1 1 91. N * & 1 & #7 V7 !& ( 1 #$ 1 1 & &X1 7 92. N & & #7 V7 ! ( 1
#$ 1 1 & & 93. N & & # 7 #7 ! ( 1
& 1 1 1 #$ 94. N
1 * & & #7 #7 ! ( 1
1 1 #$ 1 1 & 7 7 7 7 95. N & 1 # 7 # 7 ! 1
1 1 1 & & #$ 96. N & & #7 V7 ! 1
1 #$ 1 1 & 1 1 97. N # 7 # 7 1 & 1 ! 1 1 1 #$ 1 1 1 7 98. N & # 7V 7 ! 1
5 #7(* m m 1 m - #$ & (
" 99. N J & 1;- & (* m #7 ! 1
#$ 1 1 & 100. N & 1 V 7#7 ! 1 #$ a &L 1 1 1 m 5_7 J 101. N m a 1 L & m a & a & ! 1 J 5_7 #7 ( #$ m 5 1L m m m 1 102. N J m 5_ 7X! m m 5 1 #7 ! 1
&
:1
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= = @& C7 !C@MB
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= = @& !C@MB
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mL (* m !
= = @ B O!C@MB
= = @& O!C@MB
= = @ #B 7!C@MB
= = @ #B O!C@MB
1 1 1 1 N &
& & & ! 1
1 1 1 N & V 7#7
1 ! 1
103.
L1 5
7 ¤ #$
#$
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105.
7 # 7V7#7 N & & 1
! 3 7
1 1 & N & V7 #7 1 ! &
106.
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7 7 D F £
1 7 # 7V7#7 & ! 1
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108.
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&
= ' 'I@ B
¡
&
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109.
G
!
5 ! ¤ G G F F *
8.1.4. The hypergeometric function 5 F4 ((a1 , . . . , a5 ); (b1 , . . . , b4 ); z)
1.
N
1 1 # 7 #7V7
& #7 & ! (
#$
Y
)( 5 75_7
1 ¤ /) / ¢ / £ " / / $ ¢ , :1!,
m (* V( 1 ! 7 (:7] ( 1 ( m (:7 ( 1 L ( m (:7X! J (:7 ! ( m ( 1 L J
2.
N
N
6.
7.
8.
N
N
N
7R@CB
@CB
7 X@CB
N
5.
4.
#$
N
( : ( 7 ( 8 ( 7! ( m ( ( 1 L ( ( 1 L J ( m ( (8 7! J
= ( m ( ! m 5_7 m ( (:7] m (* 7 mJ (:7 m 5 m ( (* m ( ! (:7 L "!& m 5 ! m ( (* ! m ( (:7X! m ! m ( ( = m ( ! * (
! m 1L J
m m ] m 5n (:7 7 mJ 5n m 5n ] m 5 !m 5 L ! m 5 ! m ] m 5 7 m 5n 5 ! N m 5 ! m 5n ! & J m 5n m 5n! L = m #$ m (* 1 ! 7 (:7] ( & ( m (:7 ( (:7 ( & L ( m (87! ( (:7X! ( m ( ( & L J (:7X ! ( m ( & L ( ( & L J ( m ( (: 7X! J
J
= ( m ( ! #$ ( ' m 5 1 7 ! - a 1 m 5 ( m 5_7 -
7 C7( ( m (:! 7X! ' N ( 5 ' m ] ( ] 5_( 7 m ( ( 5 7 5_ @ ! m ' &Jm m m 7 L == (' m m 5 5 1 7 #$ ! - a 1 m 5n 5_7 -
7 m 5n 7 m C5n7 ( 5_! 7X! ' N ( 5 ' m m 5n5 5_ _ 5 7 m 5: 5_! 7 L = = @ m ' &Jm m #$ ( ' (* ( ' mm 5 1 1 1 7 1 1! 1 J C&7 ( m ! L ' N ! ' & (* m
a a m
- - 1 -
&
& & &
3.
:13
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B O !C@CB
B O !C@CB #$
¤
(' m m 5
- - a 1 & &
9.
N
11.
N
7 ( m
(
-
12.
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&
#$ 1 5 1! 7
- a & 5 & ( ' (* m 5_7X! ' C7( m ! ' & N 5
1 ( 1 ! & ( m 5_ 7
N
14.
N
15.
N
16.
N
#$ 5 7 ! a 1 & ( m 5: & (* m 5 m ( (* #$ ! 1 1 ( m 5n (* m 5n
#$ ( 1;- m 7 ! ( ' m 5 a a a 1 m V7 ( m ( 7 5 m ! ' ! N 7 ( ( ' (* ' & J m m ' #7( m (*' ! m 5 L 1;- m 5 1 5 7 #$ ! - a 1 - & a & 5 & & ( 5_7X! & ' (' ( 5_7 m ! C7 ( m ! ' N & (:7 1 ( m 5_7 5n
( 1- m 5 - - a 1 - a & & & C7 ( ( m ! ! ' N m ' &
L1 5
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= = @ B O !C@CB
= = @ B !C@CB
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#$
#$ 1! 7
& 5 ( ' ( 5 (:7 #$ 1 ! = = @ & B !C@CB ( ( m& 5
( O ' V( ' ( O' V( ' '65_7 & ¢ = '! % @ & '! % ¤ ( ' V7#7 #7 7 L 3 = ' % @ = '! % @ O' ! % J ! #$ m 5_7 ! ( 7 ( 5_7! ( m 5_7X! P K _ 5 7! ( ( m 5_7X!
P( 5 7 P( 5_7P( m 5_7
a ( 7 #$ m 1 ! Y N = P( m ( ! ( 7R@CB
a P ( 5_7 &
1
V7#7 1 #$ 7 = = @& !C@MB 1 ( & & & 1 ! 1
13.
(' m 5 1 N - E-Q1 - & & & ( ' ( (* m 5n! O! ' N m ' & m (
10.
:15
17.
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5 7 m 5 1 m 5 m _ m m 5 1 m 1 5 1
Y
m 5 & m 5 1 ( ! 5 m &N m 5
1 & # 7 #7
! ( 1
18.
N
19.
N
20.
N
& #7V7 & & 1 & & & & ! (
m 5_7 m 5 1 m m m 5 m& 5
#$
1
N
5 1m 5 m& 5 !
7 #7V7#7 #$
& & & ! (& 1 &
m 5 7m 5 1 m 5 m m 5 1 & m &X5 1 1 & 1 1 5 Y N
m &
N
23.
N
N
N
25.
G
!
= = @ @CB
m (:7 m & a V7 m 5 ! 7
#$
= m
7
= @& C! @CB !
¤
1 1 #$ &X1 5 # 7 ! 7 5 1 1 & m 5
1 7
# C7 m 8 ( 7 ! O m = m
7 7
= @ C! @CB !
= = @ #7 O!C@CB
= = @ #7 O!C@CB £
$# &X1 m 5 1 # 7 m 5 1 & ! &X( 1
1 1 1
1 # 7 # 7 (
1 ! 1
1 #$
V7#7 1 & 7 & & ( & & & !
1
#$ V7#7 1 7 1
& & & ! ( 1 1 1 #$
24.
= @ C! @CB
22.
21.
= m
= = @ @CB
#$
77 £
#$ 1 #7 (& 1 & a 1 Y N & m 5 m 5
m ( m 1 #7 7 #$ 1
1 m 5 !
1 7 7 F G
#$
#7
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= = @ O!C@CB
5 7 m 5 1 m 5 1 1 m m _ N m m 5 m 5 m 5
£ a 1 X1 -21
5
#7
L1 5
#$
! 1 ( 1 m 5
( m F G m m 5 J 11L J
m 5 1 1 m 5 m ( ! m m (:7 ! & N
( m ] m 5
& V7#7 #$ 7 7 1 & & & ! ( 1
#$ " 7 ¤ 1 & #7#7 1
(1 1 7 F G ! 1 #$ V7#7
7O7 1 1 & & & ! (1 1 1
#$ 7 & #7 #7 1 & 7 1 & & & & ! ( 1
& #$ & V7#7 1 7X 7 & & & & & ! ( 1
& #7#7 & #$ 7 7 F3
1 & & & 1
! 1 #$ 7 7 & #7#7 & F3 G
& & & & ! 1
1 & V7#7 #$ 7
G F 3 & & & & ! 1 1 m 5_7 m 5 m 5 m 5 V7 #$ m ( ! 7m m m 5 & m1 5 & m1 5 & ! 1 1
m 5 m 5 #7 ! 7 #$ Y N = m m 5 1 1 m 1 5 1 &
1
1
26.
L
m 5
J #7 #$ L
27.
N
N
N N
N
= = @ !C@CB
= = @& C! @MB
N
N
= = @& O !C@MB
7 G
= = @& !C@MB
N N
= = @& !C@MB
35.
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34.
33.
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32.
31.
30.
29.
1
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28.
11 ! &
:1:
! = @ !C@ B
#$
7 # 7#7 #7 &
1 1& & & & 1
1 ! #$ & V7#7 1 & & & & ! 1
m 5_7 m 5 1 m 5 & m 5 #7 #$ mm 5 m 5 m 5 ! 1 £ - a 1 F m 5 7 G
36.
N
N
= = @& 7!C@MB
= = @& C7 7!C@MB
38.
37.
N
m ( 1 L J m m J 5 # m & m (:7! m m ( ! & N m 5
m 5_7 m 5 1 m 5 1 m 5 1 #7 m m 5 &m 5 &m 5 ! & 1
39.
N
#$
40.
41.
N
N
42.
N
43.
N
44.
N
1 1
& &
1 1 1 #$ & & 1 ! 7
7
= = @ C7O7X!C@CB
#$ ( 7! 1 1 #7 ! 7 ] m : m & N m 5 m 5 = m h7 ! = @ C7 !C@CB
D F <7 G 3 F 7 G F G 3 F < GIH ¤
7
7#7 #7V7#7 & ] ! 1
7 #7 #7#7 #$ & & & & 1 ! & #$ & V7#7 & 1 & & & ! &
1 1 ( ' 5 5 m m 1 ( ' m 5
m 5 L (J 1 L 1 # 7 & #$ ! (* m
7 ¤ 7
= = @ @CB
£ F 3
7
= = @& !C@MB G
7 a 1 5n' 'g5 7 & 1
! :1!<
#$
a J & L ' 1 J&L'
5 7L ' J &
J&L'
= = @ B !C@B
L1 ;
8.1.5. The hypergeometric function 6 F5 (a1 , . . . , a6 ; b1 , . . . , b5 ; z)
mm 5 1 5 1 a N m 5 m 5 5 a
1 1
a 1 N
& a a 1
1.
N
2.
- m a 1 & a a & & & (' ! ' 5n ! N ' & m
m N a 5_7 1
Y 3
3.
N
& C7 (
4.
a & #7V7 ( m 5n (' m m 5 1 5 - a 1 - ( ' & m 5 7! & ' (* m ! ' N &
N
5.
1! a a &
#$
m h 5 m 5 X7 !
& 1 a 5 7 a 5 1 1 N & a a a a
&
#$
5 1 7 ! & ( (*'
#$
a a & a 1 & & & 5nm ! ' ! ' m
7 !
&
#$
= @CB
m 5n')(:7 ( (*' 1 #$ ! = = @ B O!C@CB 7( (*' & ( ( '
#$ 7 7X!A ( 5_7X! ( (:m 7X!A5_ ! (:7X!A 5_7X! m ( 5n ] 3hf- ¢ 3' ¢ <3 3hfT3 £ ¤
#$ 1 5 ! 7 a & & ( 5 & (* 5: ( (:7 1 #$ ! (* m 5_7 ( 5 &
$ # (*' m 5 1 7 ! a a & ( ( ' & ( ' ( #7( ( ' 1 N & 7 ( m ( ' (:7 (* ( ! '
( ' 1 (*'
Y
( 1;a & a &
a a a
1 a 1
a a !
= = @ B O !C@CB
= = @ B C! @CB
6.
m 5_7 ! ( 7 N ( m 5_7 ( 5_7 ( 5_7 ( K5_7
( m 5_7X! (K5 7! 5 7 7 m ( ( _ 5 7! (P( m 5 7! & N ( 5_7 ( _ 5 !7 L J ( P( m 5_7X! = ( (* (* P( m 5 !
@CB
:1L
7.
m ] (* ( 7 ( 7 ( m (:7] ( (:7 ( ! (:7 (P(:7 L N J : ( 7! (P(:7X! ( m ( (87 7 ((:7X! : ( (P(:7X! & N ( m (:7 ( (:! 7 L J = ( (#! 7 ! ( m ( ( (#! @CB W W 5 ] #7 (* a m 1 - a a a 1 a a a #$ m m 1 m W 1 W 1 N m 5 m 5 a - a a a a a a ! ( 1 & 1 !& 1 5 J m & L = = @ B !C@CB m 5n ! # 1 1 & & & - 1 1 & a 1 1 $ 7 7 2 F G = @ B N 7 a
1 & 1 1;- 1 1 1 1 1 1 ! 1
W W #$ m m #7 ( m 1 a <- a a a a N 7 a a a 1 W a a a 1 W a
& &-
& ! 1
1
1 5 m & L @ V B !C@CB J m 5n O! m 5_7X! = =
8.
9.
10.
1 1 & & 1 11. N 7# 7 & <-
1
7#7 #7V7#7 #7 12. N & ] ] ! 1
3 ]
-
a #$ 1 1 1 a 1 1 !
7 O
7
= @ B
7 7 D F
8.1.6. The hypergeometric function 7 F6 (a1 , . . . , a7 ; b1 , . . . , b6 ; z)
1.
N
( ' m 5_7#7( m ( 5:'g5 1 7 ! - 5_7 a a 1 & m (* 5 7 m 5n '65_7 ( m (* ' &
5_7X! ' a a 1 ( L ' - ( 5 7 L ' m J
5_J 7 a a 1 L ' m ( 5_7X! ' ' L J
J
:1P
#$
= = @ C7B !C@CB
N
2.
N
3.
( ' ( m
m (* 5 & O'! % m £ - ' % 5 1 J
m 7( m 1 1 1 a 1 m 5 1 m 5 1 7 7 a 1
&
L1 :
5_7 m ( 5 1 m ( 7 & m ( 5 7 a 1 (*')( 5_7X! ' 5 1;- L ' m ( J
L ' m ( 5_7X! ' J m ( 5
5n O'65 5_7 7 #$ ! S #7<5 m 5n'65 S
( 5_7X! ' 7 L ' 1- L ' l S J #7 @& C7B !C@MB = ! = W # - a 1 a 1 a & - a 1 $ W
a 1 a 1 a & - a 1 ! ( 1 m 5 1 L @ # # B 7!C@CB m J m ! = =
W
W
& -
1 1
7# 7
N
4.
N
5.
1 1 1 1 - 1 a #7 #7 - 1 a ( 1 1 1 1 !
N
N
N
9.
8.
= = @ #B !C@MB
7.
# 1 1 1 & ( 1 5 1 $
& & = = @& M7 !C@CB ( 7V7#7 #7 1 ( 5 1 1 1 1
! # 1 1 1 1 1 1 ( 1 5 $ 7X
1 1 = = @ @CB 7V7#7 #7 ( 5 ( 1 7
1 1 ! 1 1 1 1 & 1 - 1 a #$ @ & C 7 !C@MB == 7 7V7# 7 # 7 1 & - 1 1 & a 1 ( 1 1 1 ! 1 # 1 & 1 - a $ = = @ @CB 7 7V7#7 #7 1 - 1 a 1
! 1 # 1 1 1 1 ( 1 ( 1 7
$ 7 7 2 IF G 7V7 1 1 ( 1 1 1 5 1 1 ! 1 = = @ O!C@CB
6.
#$
N
:,9N
N
10.
1 1 1 1
7 # 7
11.
N
1 1 1 7# 7 # 7
- & &
1 & 1& #7 1 - 1
- 1 1
a & #$ 7
7 7 F G ! 1 1 # ;a 1 1 & 1 $ 1
1 Ia 1 ! 1 1
1
a &
= @ B
= = @& C! @MB
8.1.7. The hypergeometric function 8 F7 (a1 , . . . , a8 ; b1 , . . . , b7 ; z)
( ' ( ' ( 1 m m 5 1 5 1 V( m (
' V( (* ' 1 ( 1 V( m (* ' 1 ( m (*
N
1.
7 # 7#7 # 7 & - & a & #$ 7 <3
- & a & ! 7
5_7 a 1 a a 1 a
! - a 1 - & a & & - 5_& 7 & - a & 1 - & a
N
2.
N
3.
&
1.
1 N
#$
& 5 7! ' ¤ 5 7!
'
( 7
- 5_7 #$
!
( @CB
8.1.8. The hypergeometric function
1! 7
¤
5.
m ( 5_7X! m ( m 5_X7 ! m ( (
&
& & & #$ & 5 7 ! & 5 7! & N & !a = m ( ( 1
& #7#7 #7 #7 1 - 1 1 ;a 1 #$ N 1 Ia 1 ( & & & & & - 1 1 ! 1
1
1
7 #7#7 #7 #7#7 #$ 1 - 1 Ia 7 ¤ N 1 Ia 1 ( & & & & & - 1
1 1 !
4.
(*')( 1 V( m ( (*'65 ( ' m 5 5 1 m 5 5 m 5_7X! ' 5_7! ' m 5n m 5_7X! ' 5_7! ' m 5
1 1
#7 1
1 1 1 # 7# 7 # 7
1 1 1 #7 #7 - 1
10 F9 (a1 ,
. . . , a10 ; b1 , . . . , b9 ; z)
a # & 1 & $ 1 a 1 &
& ! 1 :, 1 -
= @ B
= = @
@CB
L1 P
8.1.9. The Kummer confluent hypergeometric function 1 F1 (a; b; z) 1.
2.
m 5 ! 5 ' L N1 1 J n ' ( ! 7( ! m m ( N1 1 J 5n' ! L ( S Y 43& J
" a m $9% S -21 3P& & a D - 1 N 1 J ! L H ¤ -
S ( ( 7 ! (& ' ! ' m 5n'! ( ( % !' ( m !'
m !'
$ L 0fg3 3 3 ¢ S - a D - 1 " m 5 ! 3. N 1 1 J (*' L m ! ( C7 ( ! ' ' Y $ L 0fg3O 3P& a2S D 2a S 2- 1 1 J &m ( ! 4. N 1 1 J ' (*' " L & " ( ( 7X! ! ' % C 7 ( ! $9% a -; 3P& & a2S D - a2S -21 - m ( ' 1 &5.
6.
7.
8.
7 m ' 5 ' L F fg3 G F G 1 - A 4 3 ¢ $ L N1 1 J m ! n J m (87! " 7 Y m 5n'! " OF f- 3 G 7 a m! ' & 1
* ( ' G G L F fg3 3 F A
1N 1J m J $L (* O')(:7X! 7 Y m m (*'! " " F f 3 3 G a " 5 7 S ( 7X! ¢ 1 N 1 J ! L 3 4 1A- $9% S - 43& 3 5 7 N1 1 J '65n ! L '! % 43 ¢ ¢ 4 -;&-21 S $ 5n 3P& X 43 ¢ a 3 & $ % S 1! 1N 1 1
9.
&
#$
:F G A /
:, ,
F )
& G
,
F )
m N 1 J ! L H ¤ m N 1 J ! L H ¤ m N 1 J ! L H ¤
a -21 F G ¤
¤ F -;&-21 , G ¢) & ¤ - a ;- &-21 & 0 ¤
& G ,
¤
#$ ! £ - & F G A
1N 1
& Y 3 1 & F G & , 3 £ & / F G & , ) ) & £ ¤ & 1 & F G , & F G & , ) )
10.
8.1.10. The Tricomi confluent hypergeometric function Ψ(a; b; z) 1.
m 5
J
( 7! ' ' % " "'%& L m ! ( m ( 5_7X! ( " Y $9% - -; a 1 & 9 a2S D 2a S 2- 1 m ! L H ¤ J &-
5:' !
2.
¢ 58' ! L 3 SEa 3 f S
J
m (
S " m! ¤ a Y ( * 5 ' ! ED - J L H " J $ L m m ( ! ' "!& L m ! ( m ( 5_7X! ( m ( 5 5_7X! ' ' " Y $ L ( m (*'65_7X! " a2S D a2S -21 m ! L H ¤ J J ( 7! ( &(' % ']" & S " m! ¤ a m ( 5_7X! ' H 9 $ % L & 9 D S 2 1 J L
3.
J
m 5
( '!
4.
J
m (
( '!
m! ¢ ¢ = '8>?7A@ B mJ 5:' L 3 &-21 3 1A- ;- &1A--2 1 -; & m! ¢ ( ' 6. J m L 3 7 Y ¢ 3_f & -;&-21 3& 23 1A- $ a -;&-21 3P& 1A- - 0& ¤ &2- 1 1 m ! ( 7! ' 5n' L 7. 1 - -; A
J m ( ! ¤ Y -;& - a 1 & a
2 1 % G 3 L F F
2- 1 G J & 5.
:,93
+
m 'g5 ! 1 £ 43& -; A
( 7X! " Y $9% -;& - a 1 0& & J -21 £ £ Y 3 ¢ /
& l -
$
L1 N
L ¢ h 3 fI - P5 ! % Q( ! ]& - £ If
£ ¤ / - -
Y
m !(*' m 5 1 1
J L' £ Y 3 ¢ /
l -
10.
'
7 G L F J & £ & - -21 % P 5 ( ! ! #& - £ If
5n7 7 ( ! ( 1A- 3 !¢ & S1A L J ! S 3
11.
5_7 'g5 7! L 3 J Y 3P& - a2 S
7
Y
12.
m ! ( 7! ' ' % (*' L C 7 ( m ! ' 1 - A
J m " ( ! a $ ¤ Y
; & 2 1 G F3 J L - a -21 F G $9% &
8.
9.
1
£ ¤ / - -
43& 7 $ S - - a 1 3P&
' % ! J $ L 4 3P& a2S (] ! - S 43 & - 2- 1 0& 43& a2 1 :,!5
- 2- 1 0& ¤ -21
'I@CB = ?
5 '65 Y /
13.
1!
1
#$
( "' ' % (! ]% ! $
/ - a2S -21 3P&
5
!% $&% -; a 1 & & a2S ( ]! £ - S -21 43& % a2 1 P5 ! ?'I@CB Y -21 % Q( ! £ / - - = / - -21 7 7 G -;&-21 43& F SEa 1 (*' L ( & ' J
SE a 7 a 3 1 $ SE-;a &&-2- 1 3P& & -2- 1 a 1 & ¤ 1 " ( S 3 ¢ S £ 1A-; S -; J ' ! " L $ L J a 5n'65 ! '8>?7A@ B Y &-21 % $ $ 5n')( ! £ / - a 0 / = & 2 1
14.
1 ( !
'65 1
15.
5 1 ! 1 (* '
#$
#$
#$ 1 ( ! 16.
1 (*'
( &(' S '! % ( 7! ' % % $ 5n 9 $ % S -21 0 & X / 0 / a ( ! -21 5 ! £ £
; & 2 1 % a 3P& % ( ! / - &1
17.
1! 1 &
#$
£ & V 1 / A
- a ;- &-21 43& - -21 / 0 = _'I@ B
& ¤ F G , )
8.1.11. The hypergeometric function 1 F2 (a1 ; b1 , b2 ; z) 1.
7 ! 1N 5 1
(8 7 £ / N (: 7 ! 3 £ K3 ¢ 1 1
:, ;
( 1 £ / N
5 1
1
!
&
#$
¤
+
L1
( #$ 1 £ £ ! 1 N 5 ! 7 & / 1 N 5 ¤ 2. / 1 1
" " " C7( ! 3. Y 3 £ O - 9- £ 3 £ 3 £ / <3 a £ 3 £ £ / £ @ B £ K / / 0 = m ! 4. N 1 m 5_7 1
7 7 £ £ / N ¤ ! ! 5 _ 5 7 3 / N 5 5 7 _ 5 7 m m m m m
1 1 1 &
m ! 5. N 1 m 5_7 &
7 7 7 (:7 m £ / N £ ! ! 5_7 ¤
3 / N 5 5 m 1 m m 1 1 m 1 m
5_7 ! 7 ( ! 3 ¢ 6. N L F 3 3 G ( % 1 & J 1 J L(
a2S
£ 5 Y 3 ¢ $ L F 3 G I+ - a 1 . L A- !- &4 £ / ¤ J H - a2S - -!&4 / 5_7 S &L ( & ¢ ! J % % 3 $ L 3 7. N 1 5 J
£ Y 3 ¢ - L 3 3 ¢ I+ - a -21 . L - - - £ / J H / ( 7! ( - 5_- 7X! - ¤ 3 $ (* 5 7! ( (:7X! - -21
7 (* /
8.
) ' ( 1 N1 '65_7 O! '65_7 '& '65_ ' 7 % ! 1 -; / 3 /
&-21 / 3 &-21 / / &-21 / &-21 / / &- / 0/ 7 (* ' 0/ ¤ &- / / 0 / :,9:
9.
1N
Y
10.
1N
Y
11.
1N
Y
#$ " A")" S 5 1! ' % % F 7 G ¢
SEa 1 43 J $ L 1 ( ( $ L ' 5 5 & '65_7 J
a
£
-21 ¢ a 7 a . a / 1 G F 3 3 I +
- -21 £ a / 1 £ / a 7 ¢ £ 3 F 3 3 G a ) £ /
£ £ / L1 ££ / <3 1 £ £ / L £ £ / / H / / H / 1 £ a -21 1 ¢ 3 a a 1 £ / / ,
1 £ 7 a . & -21 ¢ a 3 F 3 3 G I+ - -21 a a 1 £ / / 1 #$ 5 S 5 1! J 1 L % ( & F 7 G 3 ¢ $ L J 5 5 ]'65 1
7 ¢ a 3 ¢ i£ - - -21
- - -21 (1 ( $ L ' J £ £ £ 7 Y / £ 1 £ / 1 £ / / &4 / a 5 : ( X 7 ! $ % % Y -21 5 $ ( Q(87! % % ¢ & - a £ -21 5 $ (87! % % / 3 £ / 5 $ ( Q( ! % % ¢ & - -21 £ . & -21 7 3 F 3 3 G - + a 1 a 1 £ / a 1 / #$ 5 S 5 &! J & L % ( & F 7 G 3 ¢ $ L J 5 5 ]'65 1
7 ¢ a ¢ i£ - - -
- - !- &4 ( ( $ ( 1 L ' J £ £ £ 7 Y / 1 £ <3 £ / 1 £ / / 3 &4 E / :, <
¤
¤
+
12.
13.
1N
1N
a 5 $ 5 7! % % Y 5 $ ( 5_7X! % % ¢ & - a £ 5 $ 5 7! % % 2 1 / ¢ & - -21 5 ( Q(:7X! % % $ 3 £ / £ . 7 & -21 3 F]3 3 3 G - + a 1 a 1 £ / a 1 / #$ 5 S 5 1! J 1 L % ( & F 7 3 G 3 ¢ $ L J 5 5 1 (*'
£ 7 Y & -21 F 3 3 3 G I+ &- -21 . -; a 1 £ / -; a 1 / ¢ 3 F G £ / a -21 5 $ (:7X! % % 7 $ ( (:7X! % % ¢ & - Y 5 £ / a £ -21 5 $ (87! % % / 3 £ / 5 $ ( Q( ! % % ¢ & - -21 a ¢ 3 ¢ i£ - - -21 O - - -21 £ £ Y / £ 1 £ / 1 &4 / #$ 5 S 5 &! J & L % ( & F 7 3 G 3 ¢ $ L J 5 5 1 (*'
£ 7 Y & -21 F3 3 3 3 G I+ &- -21 . -; a 1 £ / -; a 1 / ¢ 43 F G £ / a 5 $ 5 7! % % 7 $ ( P5_7X! % % ¢ & - Y 5 £ / a £ -21 5 $ 5_7X! % % / 3 £ / 5 $ ( (:7! % % ¢ & - -21 a ¢ ¢ i£ - - - O - - !- &4 £ £ Y / 1 £ <3 £ / 1 3 &4 E / :,9L
L1
¤
¤
¤
14.
! 1 N m ( 5 1 ] m ( ' 1
J
a2S ' 7 Y $ L F 3hfg3 G J &- Y
m
15.
16.
5 1 ¢ SEa ! 1 N m (*' ( m 3 S Y ¢ $ L F fg3 3 J
m 0 . N1 J m ( ! ' L O I+ 1A-
m 5 1L ( m L J ( C7 * ( m !' $ 5 ¢ J L F a -21
L J a -21
F G -21
7
3hf G 2a S F G / a - -21 / ¤ / a - -21 0/ C7( m ! % m ! F G 1 L ( C7 ( m ! ' J
7 a ¢ a $ 5n' 3 G S F G J L Y a - - a 1 / - -21 / ¤ a - - a 1 / - -21 / £ ¢ ' " m (*'! " a -21 £ / ¤ $ L a -21 / J
8.1.12. The hypergeometric function 2 F2 (a1 , a2 ; b1 , b2 ; z) 1.
2.
m 5_7 m N J m ! a 1
N 'g 5 7 # 7
m ( : 5 (* m : ( 7 N J m ( 5_7 ( L ! 5_7 ' A J $ L F G ¤ ! L
b ¢L£ 0 ¤
8.1.13. The hypergeometric function 2 F3 (a1 , a2 ; b1 , b2 , b3 ; z)
m m 5 1 ! #$
N & 5 1 1
m ( £ /
1.
2.
( m ( 5 ( 5 m 1 £ / N 5 1
&
7 ( m! m! N ¤ N ` 1 1^ 1 1^ ` £ / N &
:,9P
m 5 1 ! #$ 1 1
( m 5 7 ! 5_7 &
#$
= = @ C7 !C@CB
+
L1 3
( ( 5 m m 5 1 ! #$ (87 m m 1 ! #$ £ m (:7X! / N & ( 1 1
N & 5 1 &
( 5 ( 5_7 #$
m ( ! m 1 m ! £ m (:7 / N & = = @ C7 !C@CB 5 1 & 7 m (: 7 ( ¤ (:(:7X7! N m (: ! ! (:7 ` m 1 1 ^ (:7 ` 3 1 N 1 ^ #$ ( ' ]'65 ' % 1! ¤ /
' ! H G % D F V N &
1 1 &
F / # G
#$ ( ' ]'65 7 ' % & !
/
6 ' _ 5 X 7 ! % H a 1 F # G a 1 F / V G ¤ D & &
N & ' '65 1!
N & '65_7'65_7 '65_7 F3 G ¢ / 3 ¢ 0/ <3 £ 3 ¢ 0/ ¤
3.
4.
5.
6.
7.
1 1! 8. N
& 7 #7 # 7 #$ 1 1! 9. N
& 7V7 &
#$ 1 1! 10. N
& & & &
#7 #$ 1 ! 11. N
& & & &
#7 #$ 1 ! 12. N
& & &
#7 #$ 1 ! 13. N
& & ]
7
7 ¤ M E( ! # / E
Z
7
7
Z
Z
/ E ¤
!
¤
Z
7
7
L
£ / E ¤
2!(
!95_7 ¤
7 ¢ VQ3 ¢ £ / 3 £ / 1 £ / £ £ £ £ £ ¤ K / L1 / 3 1 / L / 0 : 3 N
7V7 N & J ! L 7#7 !
N & & & &
7 #7 !
N & & &
7V7 !
N & & ]
14.
15.
16.
17.
7
!(:7
Z
7
Z
¤
7 £ ¤ M E( ! L / E
Z
!(:7 ¤
(
!(
!(
C¤
8.1.14. The hypergeometric function 3 F0 (a1 , a2 , a3 ; z)
( ' 1 # 7 ` ( ' #7 N ( 1 & &N ^
1. 2.
/ I+ &-21 . D ;- &-21 F / + &-21 . D H a F 1
a 1 F IG H ¤ ¤ G 3 a 1 F GIH G 3
L
8.1.15. The hypergeometric function 5 F0 (a1 , a2 , . . . , a5 ; z)
( # 7 . N ^ 1 - 1 & ` £ -;&-21 / + &-21 Y H a # -21 E3 L a # 2- 1 ' 1
1
-;&-21 V -21 3 a 1 V 2- 1 ¤ . ( 1- & #7 +
& ! & N / 8F 7
1.
2.
8.1.16. The hypergeometric function 4 F1 (a1 , . . . , a4 ; b1 ; z)
N 1
1.
2.
N 1
( ' ]'65_7 m #7( m 1! 7 7 D a F G F -;&-21 ( ' ]'65_7 m #7( m 1! ( 7
3 ¢ D a F G -;&-21 F : 31
7
7 7 ¤ G G
F a F GIH ; - &-21 7
G 3
7
7 ¤ G a F G9H -;&-21 F
+
( ' 1 (*' 1 (*' & ( N 1 1 (* ' !
( ' ( (*' V( (*' 1 1 N 1
( 1 ( O' !
43 ¢
'
3.
4.
#$
F3
1 (*'
a 1 F
G
) * ,
L1 !<
¤
) * ,
#$
a G 1 a 1 * , a 1 * , ¤ ) )
8.1.17. The hypergeometric function 6 F1 (a1 , . . . , a6 ; b1 ; z)
( ;1 - a 1 5 7 1 & #$ N 1
1 ! (
£ £ £ £ I a 1 -21 K3 a 1 -21 a 1 -21 3 a 1 £ -21 -;&-21 £ -21 3 £ -;&-21 £ -21 -;&-21 £ -21 K3 -;&-21 £ -21 ¤ ( #$ 1;- a & 5: & 3 '65_7X!A '65n O!0 N 1
& ! ( Y a £ -21 ' £ a £ -21 a £ -21 &4
&4
&4
3 a &4 £ -21 -;&!- &4 £ -21 £ £ £ £ ¤ -;&!- &4 -21 -;&!- &4 -21 K3
-;&!- &4 -21
1.
2.
8.1.18. The hypergeometric function 8 F3 (a1 , . . . , a8 ; b1 , b2 , b3 ; z)
( 1;- - <& - a 1 a a & 5_7 #$ ( 1 1 &! ' Y D / £ -21 F 3 / £ -21 G ' / £ -21 F 3 / £ -21 GIH -;&-21 £ -21 ' £ £ D / -21 F 3 / -21 G ' £ £ £ -21 / -21 F 3 / -21 G9H
; & 2 1 ' 3 D / £ -21 F / £ -21 G ' £ £ £ / -21 F / -21 GIH a -21 1 : 3,
1.
N &
3
' / £ -21 F / £ -21 G ' 3 / £ -21 F / £ -21 G a 1 £ -21 H ¤ #$ ( 1;- - &<- a a a 5n N & & & ! (
'65_7X!A '65n !A '65 !A '65 ! ' Y D / £ -21 F 3 / £ -21 G ' 3 / £ -21 F 3 / £ -21 GIH -;&- £ -21 ' £ £ D / -21 F 3 / -21 G ' £ £ £ / -21 F 3 / -21 GIH -;&- -21 ' £ £ D / -21 F / -21 G ' 3 / £ -21 F / £ -21 GIH a £ -21 ' 3 D / £ -21 F / £ -21 G ' £ £ £ ¤ / -21 F / -21 G a -21 H D
2.
8.1.19. The hypergeometric function 0 F3 (b1 , b2 , b3 ; z)
1. N & 1 1 1 1 £ &4 1 £ £ &4 1 3 £ £ &4 1 <3 £ &4 1 1 £ £ £ £ £ &4 1 £ &4 1 1 1 &4 1 &4 1 £ £ £ £ £ &4 1 <3 £ &4 1 1 &4 1 1 &4 1 <3 £ £ £ £ 1 £ &4 1 1 £ &4 1 ¤ ? 1 &4 1 1 &4 1 ! 1 1 ( 2. N & 1 1 1 3 F G 1 £ £ &4
1 £ &4
1 K3 £ £ &4 O 1 £ &4
1 1 £ 1 £ £ £ ¤ &4
1 1 &4
1 1 &4 O 1 &4
1 ( 3. N & 1 1 &
7 £ £ 1 &4 1 1 £ &4 1 E3 1 £ &4 1 1 £ &4 1 ¤ : 3 3
+ 4.
( " N & 1
5.
L1 ,9N
3 £ 1 & & 1 £ 1 & & 1 3 £ 1 & & 1 3 £ 1 & & 1
7.
N &
(
¤
N &
N &
N &
¤
8.
L J & £ 1 & & 1 E3 / £ 1 & & 1 & & Y / 3 £ 1 & & 1 E3 3 £ 1 & & 1 & -21 & £ &4
1 1 & £ &4
1 & & £ £ -21 & &4 1 1 & &4 1 ( " £ 1 & & 1 3 £ 1 & & 1 & & & 3 £ 1 & & 1 3 £ 1 & & 1 ( & & &
7 £ £ £ £ £ &4 O 1 &4
1 &4
1 &4
1
6.
¤
¤
8.1.20. The hypergeometric function 0 F7 (b1 , . . . , b7 ; z) 1.
N
( 1 1 & 1 & Y F / £ 1 £ F / 1
2.
N
F / £ 1
F / G F Y F / £ 1
G
G
£ 1 / £ 1 G
F / £ 1
G
G
G
F / £ 1
¤ G
(
& 1 1 & 7X Y F / £ 1 G F / £ 1 G Y F / £ 1 G F / £ 1 G 3 F / £ 1 G F / £ 1 G Y F / £ 1 G F / £ 1
: 395
G ¤
¤
8.1.21. The hypergeometric function 2 F5 (a1 , a2 ; b1 , . . . , b5 ; z)
1.
N m Y
2.
N m
3.
N m Y
4.
N m
5.
N 1
6.
N &
#$ mm 5 1! 7 F £ fT3 7 G :F £ f- 7 G 1 - 5 1 m 5 & ] m ( 1 ] m 1
£ £ £ £ ¤ 1 1 ' 1
-!&4
-21
-!&4
-21 1 #$ m m 5 1 ! ( F £ fg3 7 G F £ f 7 G 1 - 5 1 m 5 & ] m ( 1 ] m 1 £ Y f £ 1
£ -!&4
£ -21 1 3 -!&4 1 -21 1 £ £ f -!&4 1 -21 1 £ £ @ C7 !C@MB !- &4 1 -21 1 = = #$ mm 5 1!
7 m ( 1 & 5 1 m 5 & ] m (8 7 7 (87! F £ fT 3 7 G F £ G 1 - f m £ £ £ £ ¤
- 1 -!&4 1 E3 - 1 -!&4 1 #$ m m 5 1 ! ( 5 1 m 5 & ] m (87 m ( 1 & # 7 (87! F £ fT 3 7 G F £ f- 7 G 1 - m £ £ Y f
£ - 1 £ !- &4 1 3 - 1 !- &4 1 3 f - £ 1 !- &4 £ 1 £ £ @ C 7 !C@MB
- 1 !- &4 1 = = #$ 1 &! ( £ f £ f ( m V7 ( m m 5 1 m 5_7 1 Y £ 1 £ 1 E3 £ 1 £ 1 ¤
- - # $ & ! ( f ¢ 3 #f9
£ f 4 -21
( m V7 ( m m 5 7 m 5 & & Y £ 1 £ 1 £ 1 £ 1 ¤
- : 3; -
+
L1 , ,
8.1.22. The hypergeometric function 4 F7 (a1 , . . . , a4 ; b1 , . . . , b7 ; z)
#$ mm 5 1m 5 1m 5 &! N m m 5 5 m ( 5 m ( 5_7 1 1 1 1
7 £
£ ¢
O Vfg3 9 1 - £ Y £ 1 ' £ 1 £ 1 1
-21
-21 -
-
#$ mm 5 1m 5 1m 5 &! ( N m m 5 5 m ( 5 m ( 5_7 1 1 1 1
£ O Vfg3 £ ¢ & 1 - (:7X! Y m £ 1
£ 1
-21 -
£ £ 3 -21 1 - 1 m (87! £ £
-21 1 - 1 £ £ -21 1 - 1
1.
2.
N
3.
m m (
mm 5 1m 5 1m 5 &! 1 5 1 m ( 5 1 m ( 5_7 &
! # m (:(*7X ! 5_7! 1 - m 1 - -21 £ 1 K3 - £ 1
Y £
-
4.
N
= = @ C7O7X!C@CB
= = @& C7 7!C@MB
#$
£ - -21 1 @ C7 !C@CB ==
#$ mm 5 1m 5 1m 5 &! ( m m ( 1 5 1 m ( 5 1 m ( 5_7 &
! m (* 5_7!
1 - m (87 m ( ! £ Y £ 1 1
-
- -21 £ £
3 - 1 - -21 1 m ( ! £ £ 1 1
-
- -21 £ £ @ C7X O!C@MB - 1 - -21 1 = =
: 3 :
8.1.23. The generalized hypergeometric function p Fq ((ap ); (bq ); z)
4-4# $ & -21 m "' ! m R"' ! m ' 1. N ^
¤4¤¤ ¤ 1 &-21 '! 5 ! ' ! ` S U -21 S -21 ( 7X! " 7 ¤¤4¤ ' 5 m ! U m $ J ! !95 Y N m R ""' ' ! m m ' ' !95 $ $ 5_ ! 7 ^ (:7X! m ! V7 m !(:7 #7 a 1 N ^ ! ` -21 (87! a 1 N ^ !(:7 ` ! ! m
2.
`
$
(:7
L = 7 &B BB@CB
3 ¢ ¤
3.
m ! !& a 1 N ^ ! ! ` !&
!
a 1 m ! 1
-21
a 7<5 m " (n ! m " m " ! 1 Y 7<5 m " (n m ! !& ! ( m " ! f 3hf a 1 N 1 = m ! !& m m &B BB m "" m "'& 9BB B m !& ! ! @ B m ! 5n 5 B B B;5 ' ! N ! ` ^ m ! \#U " m !95 '#" ' &-21 ¤¤4¤ N !95 ']" ! ` $ % ! \#U U4 XW4 \#U4 ^ '#" $ 1 5 $ 5 BB B5 $ '#" 5 5 B BB5 ' 7 B ! 5 7 m V( 7 V( 9BB B 7 &BB B ' ( & m ! 5 5 BB B5 ' ! ! a N ` 1 ^ \]W U " m ! \#U 7 &-21 ( ! ? & - ¤¤4¤ &- ( $ % ']" 8 % ! \#U U W \#U 1 ( 5 !95 Y N !9'#5 " ']m " ' ']" ` ¤ ^ !
4.
5.
: 3<
6.
7.
8.
9.
L1 ,
" # U W \ U ¤4¤¤ & 2 1 - - -
1 J V7&B B B #7 L ! ' 3 1 m 5_7 m 5 7 &B BB m 5_7 7 " D H ¤ N m m &B B B m L J $ L m J !
m ! B B BA ' ! ! N ^ `
8.2. The Meijer Function
n Gm, p, q
8.2.1. General formulas
SEa 1Rl a 1 a 1Rl a ) f
(ap ) (bq )
z
f f 3 ¢ - ES a 1Rl a 1 , l ( 5 ! 5
) 0 f , ' ! C7 ( m ( !
1.
2.
3 43 ¢ - a m 5 ! C7 ( ( ! 1 - ( & ' % & >?7 ' m ( 7 9BB B B !m " ( ! ! ! 7] &B B B $ 7 &BB B ' ! = @& #B 7! SEa a 1Rl a a 1 f f 3 ¢ - SEa a 1Rl 0 f 1Rl ) f , l 1 ) , >?7 ' m ( V( 7 V( 9BB B B ! m " ( ! 7 &B B B ! $ 7 &B BB ' ! = @ #B O! ! f 0f 43 ¢ - SEa 1Rl 0f f S a a l a1
l 1) , , l a 1 ) >?7 ' m ( V( 7 V( 9BB B B m " ( 7 &B B B ! $ 7 &B BB ' = @ #B ! !
3.
!
: 3 L
( 5 5 ' 6
0 f £f £ E S a R 1 l - l a 1 ) O 1 ¡\ £ , £f £ £ E S a - - a 1Rl
1 ¡\ £ , l 1 ) ( 5_7 P5 5 7 ! ( * ( ' (* m 5 ( ( 'g5 7 ! = @ 7X! B
0f
SEa R1 a l ¡ l £ -
4.
f 0f S Ea R1 a l ¢ £ !¡
l
£ - - SEa 1Rl £ O 1 ¡ £ f £ , l a 1 ) £ 3 £ - - SEl a a 1Rl 1 ) £ - O 1 ¡\ f £ O , ( 5_7 ! P5 5_7X! ( ( ' ! (* m 5 ( ( 'g5 7 ! = @ 7X! B
5.
6.
SEa a R1 l \¡ 0 f , l 1 ) £ E S a
l R1 a l ) 3 £ V-
SEl a 1Ra l )
£ £ ¡ 0 f £ f £ ¢ £ f £ 3 0 f ¡\ £ £ ¢ ( P 5_7 5 ! ( m 5 Q(
¢ £ £¢ £ ¢ £ £ 3 ¢ 5_7! (*'
,
£ , ( !
( ' * ! = @ C7 O! B
8.2.2. Various Meijer G functions 1. 1 M 5 7! !¡
Y
K
7
7
*
_ 5 7
3
7 7 3 5_7 *
K
: 3 P
= M 5_7X! @CB
3.
4.
L1 , ,
1 ¡\ P3 f f 3 f f 7 £ £ £ £ ¤ / - / E3 / - /
A7 @CB 3 2 IF 7 G K 7 3 7 / ¢ 3h = 1& & ¡!¡ , * )
2.
3 / 0/ 0/ / 0/ 0 ¤ 1 1 1L ¢
¡!¡ J 3 2 l & 0 7 7 E(:7 7 7 E(:7 ( 7 O !C@CB Y K 3 3 K * = *
1 & ¢!¢!!¡
5.
6.
7.
8.
9.
10.
1 ¡\)¡\
3Pfg3
3Pfg3
f f f ) ¡ f
f f ! ¡
7 7
# 1 E 3
7
£ 1 ¤
f - 2- 1 £ / 0f a 1 f - -21 £ / f - -21
£ / ¤ £ / ¤
f 7 f
)¡ 1 - 0/ 3hf fT3 f £ £ 1 - / 1 - 0/ 0f
/ ¤
1 f f
!¡ 3Pfg3 f f f 7 C 7 m 5_7X! £ a £ &4 O 1 6D
1
- -21 &4 O 1 C7X m 5 ! £ - -21 &4 1 :95N
£
7
11.
7
7
13.
&
5 7 ! 5 7 !
£ £
a & 4 1 6D
1
- -21 &4
1 C 7X m 5_7! £ 3
- -21 &4
1 _ 5 X 7 ! m £ £ &4 1
- -2 1 m 5_ X 7 ! £ £ - -21 &4 O 1 f f ¡ f 3Pfg3 f f C7X m 5_7! £ £ a &4 1 6D
1 - -21 &4 1 C7 m 5_7X! £ 3
- -21 &4 1 _ 5 X 7 ! m £ 3 £
&4
1
-21
_ 5 X 7 ! m £ 3 £ - C7 m -2 5 1 7! &4
1 H £ £ a &4
1 QD
1 - -21 &4
1 C7X m 5_7! £
- -21 &4
1 _ 5 7 ! m £ £
&4
1 - -2 1 m 5_ X 7 ! £ 3 £ - -21 &4 1 F3 / & G ¤ ¡
¡ &
&
( 5 5 ' 6
£
- -21 4&
1 £ - C7 m -2 5 1 7! 4&
1 H
m m
12.
14.
3 £
f f-
7 F G 1 &
¡ !¡\ !¡!¡
3 & ) *
,
3 & ) *
,
H¤
H¤
¤
h 3 f 3Pf ! m
:95)
# 1 K 3
£ f
V 1 ¤
15.
1 &1 ¡_ !¡
L1 , ,
43 ¢ '#" 43 ¢ ' $ L 0/ 0/ ¤ J &
¢
1 &1 3 < ! ¡ £ 3 ¢ / £ / L £ / £ / L 1 2- 1 ¢ & 2- 1 4 3 93
16.
17.
1 & 1
1 &1 ¡
1 & 1
7
!¡
18.
-
£ & 1 & &4
£ / <3
) *
&
,
) *
3 £ /
&
£ / ¢ a £ / ¤
1
,
= @CB
¤
! ¡ &4 3 1 & F G 1 & F G 1 & F G 1 & , , , , ) ) ) ) Y F G 1 & , ¢ #& 1 & F G 1 & , F G 1 & , / ¤ ) ) )
19.
20.
1 & 1
22.
! ¡
1 &
G 1 & , F ) ) Y F G 1 & , ¢ #& 1 & F G 1 & , F G 1 & , 3 / ¤ ) ) ) ¢! '%& "
&1 ¢)£ ¢)!¡ O'65_7X! D F ,G 3 a 1 F ,G9H '& ' % ¤ O'65 7! )¢ f " m ( 5_7 m ( ¤ - f ( 7 ( H
&1 ) ¡\ f , P L 3 D m J J ! L !
&4 3
21.
,
F )
G 1 & ,
F )
G 1 & ,
:95,
¢)
&1 ) !¢ f!¡ , " ! ! (:7 m m ¢) 3 24. 1 < !¡
3 3 £ / £ / / £ / 23.
( 5 5 ' 6
m ( 5_7 D 3hfI J ( L3 ! / £ / H 1 3 <3 1 £ / H1 £ / <3
¢)!¡\¢
& &1 !¡ 3 ¢ 3h& !¡\¢
1 !¡!¡\¢ &
25.
26.
£ /
27.
29.
30.
£ & -21 a £ / 1 ' >?7R@CB 1 =*
/ ¢ 3h ¤
£ / <3
£ /
£ / 0 ¤
¡ !¡ 7 £ 2 IF G 2- 1 K )¢ )
& ¢) £ !¢ ¡
28.
K
m ( 3 ¢ J ( ! L H ¤ £ /
7
7
7
7
7
7
¤ 3 * ¢ 3 K 3 * ¢ 3 ' & 3 ¢ ' 5n7 D F ,G a F ,GIH 1 ' & ' % 3 43 ¢ ' 5n7 ¤
¢! ¢
& ) ¢! )¢ !¡ , ' '#")" ' 3 ¢ C & & -21 43 ¢ ')( $ ! % ')( $ ! 1 ¢ ']")" 3 3 ')( $ ! % 43& 1 ¢) f ¢ ¢
& ) ¢! !¡ , 3hfI O 3 3hfI 3hf- ¢ ( m ( Y D ' Q3 ¢ ( ! L 3 fg3 £ 4 J J :953
= @CB m ( ! L H ¤
!¢ \¢)
& ¢)\¢)!¡\
31.
! ¡ f f !¡ 3Pf f f ¢
3fT3 0/ 2 3hf
34.
3hf
35.
3fT3 0/ 2
\¡ !¡!¡
& &&
37.
& ¡
=
0&
L1 , ,
@CB
0f - -21 / £
f a 1 / ¤
1 & ¡!¡\!
£ & V 1 & &4
F / £ # G
f - 0/ ¤
f - 0/ ¤
F]3 / £ # G ¤
" ' ' $ L / / ¤ & J
38.
&
39.
¢ 3 £ 3 ¢ £ / & 1 & ) 3 <!¡\! ¢ ,
£ ¤
2- 1 0 / f a 1 / 7 7 7 / ¢ h 3 , K) * / ¢ h 3 , ¤
f / K 7 3 ) *
!¡ f f )¡\ 3f f f £ 0/ f 0 / 3
f f 3Pf f f £ 0/ f 0/ < 3
f 3f f f ¢ f
36.
-
33.
7 C Q3
32.
¢
93 :95 5
¢ £ ¤ 4 3 /
1
40.
41.
42.
43.
1
( 5 5 ' 6
¡ \¢)!¡\ £ Y &4
1 £ 4&
1 K3 £ &4
1 £ &4
1 1 1 3 £ &4 O 1 1 £ 4& O 1 E3 1 £ &4
1 £ &4
1 ¤ & ¡ \¢)!¡\ 1 Y £ &4 1 £ &4 1 K3 £ &4 1 £ &4 1 1 1 3 £ &4
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References Monographs
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B FE # ' HGIJGIJGI !#" $! '&R(* + +, +$.-S/1-3T-PU0< V L(3O : ,=!#"'8 9 ':.;,=< >$# ?@.A 'C
B D FE # ' HGIJGIJGI !#" $W! '&X(* + +, +$.-%/1-ZY-\['! O" !3:+ ^] ! >$;_+< +5R$#`?@.A 'C
B D FE # ' HGIJGIJGI !#" $a! ' &(* + +, +$.-b/1-3c-R>d# + +$. a['! O" !3:+ X] ! >$;_+< +5R$#?@.A
'C'D FE # ' B eJf Xg h HJi j k@, l + ] ! >$ :+ +'& + !#"8>9 ':.;,=< >$Vmb/N- 2P@?@.A #e >C ''n JD ZAoQg j j; HGIJGI k@, B l + p] ! >$ :+ +'& + !#"q8>9 ' :. ;,=< >$Vmr/N-#M1q?@.A # e >C ' B n ZD JAoQg j j s 3f JA etq vue.w @k! '&x$F{|,;;l 8< +5}9 " ! $VmS~H ! O>l3$Vmv! '&\U0!#;l +5\!#;,=:V!#"v]!x " +$#>?@.A V>C> L tJ > P]Pl 4 + < @8>9 ':.;,=< 13 e eVe >j F| # . @ 3u j u u V J ]l 6 (3 O : ,=!#"v8 9 ':.;,=< >$F! '&q;l +, 6
HO3O> <J , 5\!#;,=< >$VmF/1-2P1?@.A >C @ B Jf B R Z. w N s> V . e e w# | #u 8< +5}9 " ! $S! '&F]Pl < V +5}$*_+< S;l o(3O : ,=!#" 8 9 ':.;,=< >$`<_LU0!#;l +5\!#;,=:V!#"*|l37 $V,=: $#>?@.A | Vj 'Cu w# Q | j w> S j j \h '(3O : ,=!#"*8>9 ':.;,=< >$#>?@.A >C # Zf j j ' Su + #A ee@ ;
Sn; :.;,=< z <};l }NO + !#;,=< >$S{1,;;l6(* + +, +$# ?@ .A >C @ Jj Su #e g # g #
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Index of Notations for Functions and Constants
M ]!
L @eV e J e w fK.e +e
' eV | j j fKs> ' M ! eV | j j * j f 3j M ! M ! M ! M ! M ! ; 5 M ]!
"
M ]! e+ RJj L @ eV e L J B B B @eV }h| j @ e #e
+e
( K C7! J 7 C M ! Z oe+ K Z Jj' 3 #e.w 3j M ! Z " = 7R@ oe+ w# 3j t 0 Z Jj* 3 # e.w 3j
M !
5 C Z
5
Z
oe+ K # > s j ` 3 # e.w 3j
@eV K 3 #e.w 3j Z ( L oe+ K@j 3 #e.w 3j
M ]!
M ! (
V
J
m $ ! Z ! @eV K f}>j .e`Jj j e # e.w 3j D ! :<93
#
W M ]! " ( ` oe+ ` #s j ` + j e ! ^
$ ! Z 7 ( $ @ eV K f}>j .e`Jj j e #e.w 3j eV KZ z E
' e+ Rh| j fKs> ' M ]! eV Rh| j @ j f 3j
M ]! E
Z
2(8
! @eV `bZs> +e
oe+ #e 3j #e.w 3j M ]! " Z " W ; M ! oe+ e Z " W M ]! 7( ; M ]! @Ve K f}>j JfR #e F +e
; M !
Z
Z
W @eV ` +e f w
w fR #e
" ! "" $ "
% ! ! % m " !! " Z " " C7( 4! " " C7 (4 ]! " " oe+ RD 3 S # > w# fR.e +e m ! m ! N !! ` N ^ ! ` N #40f E & ^ ! m ! " m ! " B B BX m ! " " n ¤ 4 ¤ 4 ¤ n 4 ¤ ¤ 4 ¤ ! ! B B BX ! " $9% f 1 6& N f 1 " " oe+ w# 3j t. z # > w# fK.e e " " ] ! m " $ " %e+ ^L f\fK a ! #e ! ! ! # > w# fK.e e " ] 7 B B B @eV K e 3j I e #e G " ( ' ## m ! ! ` " !( ' ## m &&B B B B BB m ` " ! ^ ## ^ ## Z % ( % ' " %$ '%& % ( & %
&
:
#
@eV }I D e !& 7 ( W #$ @ eV Rue V e ! M ]! H 5 & & & 3 8 M ]! M ]! 5 M ] ! 3 8 M ! M ;! ( M ! oeV Sg JJj e
eV e nZ z ;eV |ZJj +e eV `eV z W ' W ' M ]! ( 7X! ' ' " e+ Kg@ Vf eK j f 3j M ]! ` 5_7 ` " eV @f |ZJj ! ^ ^
e Ve eS M ]! L 5 7 ( ` @ eV |ZJj e e+ eS ! ^ J Z (* ! o e+ #w +e M ! J
K
$ ! Z oe+ ` fR>j .e`Jj j e #e.w 3j +e eS ! M ! " ! ' M ! ' M !
e+ R # 3j +e e+ Kf |ZJj +e e+ eV z M ]! M ]! M ]! M ]! ; M ! M ! eV }SJj +e M ]! " 3 '& 8 H @eV Rf zue V e L ' M ]! ' M ]! e+ B w ` j f 3j " ' -/' M ]! / ' ' & / " eV `w# 3j t. B w ` j f 3j " M ! ! " Z " = ! C7(: ]! @ e+ } j j #w e+ f eV
M ! oe+ Rh| j j #w e+ f κ 5 M ! & " κ 1 |eV S ee 3J | ! #eH # > w# fK.e
e "' ' ' M ! ' M (87! ' eV B .w# ` j f 3j Ne+ .w# e eV M ! M ! B 1;- eS
:< ;
#
M ! L J 1-
( M ! M (87! ( ( M ]! J L ! 99 L J 1;
9 ( ! ( 7! ( C7 ( ! ( ( ! 9 J 9 L
oe+ # e B .w# e e+ eS ' ' ' 3 465 78 M ! ' C7 (8 ! " 4 C7,5n ]! " 7 ' C7(8 ! 4 &(' C7,5n ]! 7 & '
M !
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!
65
5_7
565_7 " ! (65_7 9 ( L ! ( ! (
3j
= 7A@ oe+ w# 3j t 0 Z Jj* #e.w 3j '#" ( '65 $ (87 O')( " ( z ' 3 ( 8 ( X7 ! " ' ( 5 $ L ) ' ( ( $ L '#" &" eV que Vj w fKs> " J J e+ eS
Z "
M !
oe+ # e B .w# e +e `Z n M ]! Z ! @ eV K Z Jj' 3 #e.w 3j
! ! ( ( ( 7! C 7 ( ! J 9 5 7 ( 7! ( ( _
' eV ` # 3s * j f 1;- oe+ B .w# +e eV KZ z M ! a a 1 a 65 5 7 " " " M (:7! '&
& 1 W 5
M ]! '%&
a a & a a " ]"']" M (:7! 1W - " "
5 !&5 ( !
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#
7
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M ! Z ! ( ! @eV R # > s j K # e.w 3j Z ! M ! @Ve ` #e.w 3j Z ! M ]! M !( ( @eV K #e.w 3j
&
0eV %@ Zs3 V j f 3j eV ' M ! ' ! eS 1 1;- 'g 5?7X! ReV 6@ Zs3 V j Q ! ' M ]! & 1;- f 3j eV K Z z
& " 1 M ] ! R + e n ee 3J } ! #e6 # > w# fK.e κ κ5
e
" M ]! = 'I@ ' M ]! ' M ! ! oe+ K?@J f +e e+ |ZJj e e+ `Z n ( @eV Ks>.e +e ! =
" " C7 ( 4! KeV fR>j .ens>.e +e M ! Z R" " oe+ w3f\f e ] ]! Z " " @eV K f}>j JfR #e fR>j .eKw3f}f e !
Z
# !
e = m !C@ "
!E(
# ! Z " "
m " !
g $ m ! g $ & m !0! ( 5 ' 7
&B BB &B BB ' oe+ R Z J o fKs j '
:< <
e+
f}>j .e
w3f\f
#
M ! ( @eV R Jf Ft .e e " M ! @eV Rg A ett..e +e " ! 7 > @eV Kg@ +e " m m ] ]! " $ " "
"'& !! "5 " ] ]! $ " " "
m "'& !! "5 ( ( )" " ' '( " ( 7X! J $ L $ e+ Kue Vj w fKs> eV KZ z " M ! ! " "'& $ " " m # ! !! " 5 $ "'&
" ] ]! $ "" "'& !! "5 " # ! " " "
!! " 5 "'& m ]! ! ! ! $ = " " 5 ! oe+ f ' ! #e # > w# fK.e e M ]! K ! !( 9 M ]! !( D !C@ K 9 =K 9 9 !;( M ]! !( D !C@ K 9 =K ! 7 (* !( 7 ( 5 9 9 D K
] ]! ! ! "5 m ] ! ! ! "5 K M ]! = M ]!C@
m
"'& $ " " " "
" "'& " "
@eV K +e :<9L
Index of Notations for Symbols m ! m m 9BB B m m ! m m &B B B m m ( ! m ( m ! ( &B B B m ( m !95 m 5 m 5 &B B B m 5 m !95 m 5 &B BB m 5 m ! ( m m ( m &B B B m R" ( m m ! & ( m &B B B m ( m m ! ( m m ( m 9BB B m R" ( m m & ( m &BB B m ( m m ! " m m 5 7!IBB B m 5 $ (?7! m ! 7 IeV
3f}fR fKs j ' % 7 R B B B ')(87!0' C7! ' % 7 % ( 7X! % 7 '! % % B BB O')( O!0 ' ' ' % '& '65 L L ' ' '65_7X! % % 7 B B BX O'65_7X! J J
' $ ' % % $ ! % % %% ( 7 ! 7 % % $ 5_7X! % % ' $ 5_7 ' = $
" ( " ' 7 L $J L J fK m m = @ ' = ' '65_7,' 7 9BB B @ @eV #e.w# @ e / /
f f 1 4 f O f O 1 4 ¤ f S f SEa 1 ¤¤ f S f ` ¤4¤¤ f f f 9 S f9SEa 1 S
:<9P
# m !" m !" 0 m 9! 5 ! " m 5 ! " ' m " m ( m ( & B BB m ' " ( 7 ' m " M ! m " M ! ' " " ' m " m ( 5 m ( & 5 BB B5 m ' " ( ' m " M ! ' m " M ! " "
: L N
