Special Functions of Mathematical Physics and Chemistry (LMT)

SPECIAL FUNCTJO;\,S OF MATHE:\IATICAL PHYSICS AND CII:EMISTRY

M:\'I'III~;MI\TICJ\L

UNIVElls!'ry

'I'I;:XTS

AT.l~X,\~Dlm

DANIEl.

C. AITKEN, D.Se" F.Il.S. Eo nU'I'JrEIlFOHD, D.Se" On. :'IATll.

D"TI!UMISANTII ANn i\IATlIICI!S STATllIT/CAI. :'IATlllmATICS

W,Wt;8 EU:CTIIICITY 1'/t().JI!C'1·/V'-: GI':O)lHT/lY

1:o."Tl'.G/lAT'ON 1'.\/11'110/. DII"'P./I/!STI.\TIOS

ISFlslTl.: SIWII!S,

Prof. A. C. Aitken, n,Se" 1'.11.5, I'rof. A. C. Aitken, D.Se.• F.rI.S, Prof. C, A, Coulson, D.Se., F.n.S, Prof. C. A, COllison, D.Se., F.II.S. 'I'. K Faulkner, Ph.D, Il. P. Gillcspie, Ph.D. n. P. Gillespie, Ph,D. ProF. ,1. :'1. IlyHloll, IJ.Se.

1:o."TEGlIolTIO:S 01' OllDINAII\' DlI'I'l!.H1!:o.'·IM. ~UAT/OSS ISTllOllUCTIOS

E. L. Ince, D.Se. '1"0 "IlI! 'I'llI!OILY 0/' I'IS/TI! Gnoups

\\'. Letlcr"mllll,

Ph.D .•

D.Se,

ASAL\'T/C'\L CI!OMlITRY 01' TURI!I! DUlr.SSIOSS

ProF. W. I-I. M'eren, Ph.D., F.Il.S. FUSCTIOSS 01' A emll'l.!!X VAR/AIII.I-:

D. E.

CUSS/CAl. :\II!ClIASICS • V"CTOll MCTIiOIll!

•

Vor.u~,,: AS/) h'TIWIIAI.

E. G.

Phillips,

I\I.A., M.Se.

nutherrortl, D.Se., Dr. :'lalh.

D. Eo Illllherrortl, D.Se., Dr. MUlh. .pror. W, W. Ilogosillski, Ph.D,

S/'/!CIM, FUSC'flOSS Ot· :.IATllE)IATICAI. I'U\'SICS .\SI) CllI!)IlSTlLY Prof. I. N. Sneddon, :'1.:\., n,Se,

ll"rry Spain, B.A., .\I.Sc., I'h.D. l'roF. II. W. Turnuull, F.ltS.

Tlmony 01' EQUATIOSS

hi

f~tqHlrlllion

TJIlwny 01" OUmNAUY DIl'I'ImENTIAI. EQU.... I'IO.'1S J. C. Butkill, Se.D.,

F.RS.

GEluI,\s·Escl.ISII :'IATIll!~IATICAI. VOCA .. UI .... IlY

S. Macintyre, M.A., I'h.D. TOl'OI.OO\·

•

K

,\1. PUllcrson, Ph.D.

SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS AND CHEMISTRY fly

IAN N. SNEDDON .'1.:\., D.Se. 1'f10FF.$son 01' MATUf:!ltATICS IN

TilE USl\'RHSITY COU.EG!: 0.' !'\ORTII STArrOROSlllRI:

OLIVER AND nOYD EDINllUUCIi AND LONDON NEW YOIlK: INTERSCIENCE PUlll.lSIIEIlS, INC.

1950

FIRST EOiTlOS

tu:;o

",UNTIP III 1l0LLA1
11'."., (;11I01111<(;111<

PREFACE This book is intended primarily for the student of applied IUnthcmnlics, physics. chemistry or engineering who wishes to lise the 'special' fUllctions associated with the names of Legendre, Bessel, Ilcrmitc and Lngucrrc. It !Iims nt providing in II (.:ompncl form most of lhe properlics of these fuuctions which arise most fl'C
PREFACE

"

nequired a taste for the subject, and it will be obvious to anyone who knows his published wrilings how much I have been influenced by them. I\EI-:LE, ST....l-·FonDsfllRE. ~Oth

August, 1055

CONTENTS CIlAI"I'lm [

I:\TIlOI)UC'fIO:-;

,

rACE:

1. The OriJ::ill of Spttinl

Funclion~

2. Onlil"'!)' I'oilllll or n I.incir Differential 1-AllIlllion

3. Hegullir Singular I'ointa 01. 'nlc Point nt Infinity 6. The Gamma I"unction nod Helaled FlIllcliolls Exnnlplell

.,

G

o

,".,

CIIA1'Tf.1t II

IIYPEHGE0:lIE'I'IIIC

FlJ~CTIO;';S

O. 7. 8. D.

The lIypcrgt.'OlIIctric SeriC!l All l!Lltgml !,'ormllill for the ll)'fKlrgcomclrie ScriCli The 1-1)'llergeomctric E'COIllCtriC Series Examl'lC!l

18 ~O

::3 ::8 31 32 3G aD

enAl.,."n III LEGI~NDI1E

1:1. '-I. Hi. 10. 17. 18. 10. 20.

I'UNC'l'!ONS

Legendre J'olYliominls Recurrence Ilclntloll!l for the t.egcndrc l'ol)'lIominLi The ),'ornlUlnc of MU'llhy IIlId Hotlcrigues Series of Legendre J'olynominlt J~'Cldre', Differentinl Equation XeuTO.'lnn·s Fonnuln for the 1..I:'~ndre Functions Hecurrenee HeinliOlu for the Funetion Q.I}l) The UIiC of [.('uendre Function, ill Potential 'I'hear:)"


vUI

CONTENTS

P"Cr. 21. I.c~cndre·s :\'<SOCinled FunctiOIlJ ,:I 22. Inlel.'l'3l EXl'rMISion for Ih...\~i:ltcd I~gendre Function ,0 2:1. Sllrfnl)(' Sph"ri",,1 Illlrmoni~ 80 2-1. UIlC: of As-"Ot'iutcd 1.ll.... IIIIrC FUIl(,tions in Wllve Mecllllllics 11:1 Exnlllpies 85 CII"'''TKn IV

HKSSEI. FU:\,CTIOXS 25. 20, 27. 2M. 211. 30.

:\]. :12.

:la. :II. :l5. 30. :17.

The Origin of Ilcssel Funclion!! llf'C1llT'l'nee llel,"liolL~ for lhe 1IC'<Se1 COt'fficicnts St-rit:!l Expnn"ioll~ for tl,e Il cI Coefficients IIlICJ!n,1 EXllr~,illll~ fur llle III'$..~el CO("tTic;cnl'i The .\,ldition Form"l" for ll,e BC'SM'I Coc-ffici<.-nls lIrs......I'H Dirferl"lllirll F.qll:llion Sllhcril';lI ll('S.oel Funel in,u IlllcJ!mls iU\'ol\'inl:! lIesse! FUIII'I;OIl' 'I'll.., Modifi,.d H"~~el F"netinns The lIer and Iki Furll'tinn~ EXI'IfIl~ioll~ in Serif'S or 1I"H~1'1 FUlll'lion' The U~e of 11("1.<'" F"IlI'I;on~ In I'nll'nl],,1 Theory t\syrniliolil' EXI","sion5 or lles..~el Func(]oll5 EXllmr11es CIl"I~R

07 100 101 102

lOS 110 11:l

'" 110 121 12-1 127

V

TIlE FUXCTIOXS OF IIEIUIITE .\X-D 38. :.m. 40. 41. 012. 4a. +1. 45.

""

LAGUEnnl~

Ti,e lIennile l'oIynomi:'1.'I Herlllile's DiffcT\"lllial ~Al'lIIlion Hermile FUllcliorl.'l 11'e Occurftnee of lIemlilc t-'lIncliolL< in "'(I\'c )1«lmllk. Th~ LnI!IK'TTC 1'01~'nollliaI5 1.Il1.!\'~lT'l'·5 Diff~",lllial Efillatioll TII~ '\l'SOt'illled l.al!lICfT(' I'ol~'nolllials 111,,1 Functions '1,e W:\\"c Fundio05 for the lI~·tl"'J,oen Alom 1~):"ml'les

1:12 1:14 130 140 142 145 147 150 155

Al'l'KNlJIX

Tl IE DIHAC DELT,\ 40.

'I'II~

Dime Delh. Funclion

FUXC'I'IO~

150

CIlAI'TEIl. I

INTRODUCTION The Origin of Special Functions. The spccial functions of mntbcllll\t.iclIl physics arise in thc solution of partinl diffcrcntial C(l'lIltions govcrning thc beillwiour of ccrtain physical qUllntitics. Probably the most frcqucnlly occurring equntion of this type ill nil physics is Laplace's cqulll.ion 1.

(I.l

I

satisfied by a ccrtain function 'I) describing thc physical situation undcr discussion. Thc mathcmatical problem consists of finding those functions which sntisfy equation (1.1) nnd also sntisf)' cert.ain prescribed conditions Oil the surfacc.~ bounding the rcgion bcing considcrcd, For cxam pIc, if 'I' dcnotes the e1ectrostntie potcnl.iul of 11 system, lp will bc constant over nny conducting surfncc. The shapc of these boundnries oftcn mnkes it dcsirnblc to work in cur\'ilincar coordinntes ql' 1/2' 1/3 instcad of in rectangular cartcsian coordinates x, y, .:::. In this case we have relations

x ~ x{q.. q" q,l, y ~ y(q" q" q,), ,~'!q" q" q,)

{l.0l

expressing the cllrlesillll coordinnlcs in tcrms of thc Clll'vilinclll' coordinates. If CqllllliollS (1.2) lIrc such that

ox

0,1:

aq, aI/I

whcn i oF tho~onal

i

+ oy oy + az az Olf, aql

= 0

aql aI/I

we say that the coordinates (fl, If'!, q3 nre orcurvilinear coordinates. I) The clement of

1) n, E. llulhcrford, Veelor MelllOl/$, (Oliver & Boyd, Iflllll) pp. 50-0:1.

2

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

length dl is then gh'en by dl l = II; (lqi wherc

+ 1.; dq; + IIi (lq;

(1.3)

'_ (aX)' (aZ)' "t- + (a- y), +Oqf

oql

§1

(1.4)

oqj

and it cnn easily be shown that

V2VI=_,_!~(""h30'l1)+~(h:\lI,O'lI)+~(hll1" 0'11)) (l5) hl ll""3 iJ'l,

III iJql

OQ2 II" O'l2

Oq3

11 3 01J3

.

.

One method of solving Laplaee's equation consists of finding solutions of the type ~~

Q,(q,)Q,(q,IQ,(q,)

by substituting from (1.5) into (Ll). We then find that

~~(~"h30QI)+2- ~ ('. 311 1OQI) +.!.. ~ ("1 112014)_0

Q10ql

hi oq,

Q: otll

I,: oq:

Q30q3"3 Oq3

.

If, further, it so happens that

h Tt

"3 = ,

J.(ql)Ft(q,,· q3)

etc., thcn this lnst equation reduces to the form

"(," I~~!/(' )t/Ql)+p~( I L·Q3 Qtdq\ I Ii dql ~ q3,(11 ).!..-!!:...!/. (~2d(12 -. (Q.,,)dQ2) llq2


O.

Now, ill certnin circumstances. it is possible to find three functions ::1(91)' :::(9:), ::3(93) with lhe property that Jo\(Q2' 93)::1('11)

+ Pt(q:\, 9,)!::(q2) + P:\(9"

q:)g:\(q3)::::=

o.

Whcn this is so, it follows immediately that the solution of Lnplnce's cqulttion (1.1) reduces to the solution of thrce

INTRODUCTION

§1

3

self·ndjoint ordinary linear differential equations

d{ldQ,} '-d -

-/-

'h

{ql

1:,

Q= f

0 (0I = 1, 2, 3 ).

(1.0)

It is the study of differential equations of this kind which

lends to the spccinl functiolls of I11nthcmnticll.\ physics. The adjective "special" is used in this connection because here we IIfC not, as in unnlysis, concerned with the general properties of functions, but only with the propel,tics of functions which nrisc in the solution of special problems. To take

pnrticulil.r case, consider the cylindrical polar

II

coordinntcs (e, 'P. z) defilled by the equations a:= QCosrp. y= (] sin rp, z=z for which hi = 1, h~ = (!, h3 = 1. From cqulltion (1.5) we sec that, for these coordinates, Laplace's equation is of the fotm

If we now make the substitulion ( 1.8)

we find that equation (1.7) may be written in the form

((r-

1 R I d R) It d'1'!. Q dQ

+

1 (F.rj)

I

+ Q2rp llrp'J + Z

((! Z

t/z'!. =

o.

This shows that if 1>, Z, R satisfy the equations rlN) ~~ drp·

+

{r-Z (h'J -

lI'!.lJ)

~Z 'IIl-"

O.

(l.Oa)

= 0,

(LOb)

=

d'!.U+ 1 (IU+(! '1l'~' R111-_0 de'!. '1 de eZ

(l.ne)

4

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

§2

rc.~pcctivcly. then the function (1.8) is a solution of Laplace's equation (1.7). The study of thcse ordinnry differential eCJlllltiollS will lend us to thc special functions npproprinte to this coordinate system. For instnllee, eCJuntion (l.Oa) Illay be tn ken ns I. he eqllaHolI deli /ling the eirclllnr fUllctions. In this context sin (/Up) is defined as that solution of (l.Oa) whieh has value 0 when qJ = 0 nnd cos (nqJ) as thai which hng vnllle 1 when qJ = 0 nnd the properties of the functions derived thcrcCrom, cL ex . .~ below. Similnrly equation (I,Db) defines I.he exponent inl I'll llet-ions, In nctunl practice we do not proeced in this wny merely OeCl\IlSC wc have nlready cnt'ollnlered thesc functions in lInother contcxt lind from t.hcir familiar propcrties studied their relation to equat.ions (l.On) and (U1b). The situation with respC'et to equation (l,Oc) is different; we canllot cxpress ilS SOlli! ion in terms of the c1emcn tm'y fUllctions of annlysis, ns we were able to do with the other two equations. In this case we define lIew functions in terms of the solutions of this equat.ion allel by invcstigating the series solutions of tllll ellunlions der'inl the propert.ies of the fUllctions so defined. Eeluntion (l.Oe) is called nessel's equation und solutions of it nrc cnlled Bessel fUllctions. Besscl functions arc of grcnt importance in theoreticnl physics; the)' nrc discussed in Chnpter IV below.

2.

Ordinary Points of n Linear Differential Equation. \Ve shall Iin"e occasion 1.0 discuss ordinary lincnr differclltinl equations of the sccond order with varinble coefficients whose solut"ions cannot be ohtained in terms of the elcmcntary functions of mnthellllllicni nnalysis. In such CllSes one of t.he standard procedures is to dedve n pail' of linearly independent solutions in the form of infinitc series and fr'om these series 1.0 compute tables of stnndard solutions. With the aid of sueh tables thc solution Ilppropriatc to any given initial conditions may then he l'eadily fOllnd. The object of this note is to outline briefly the procedure to be followed in these installccs; for proofs of thc theorems

INTRODUCTION

s

quoted the reader is referred to the standard textbooks, 1) A function is callcd nnnlylic at a point if it i.~ possible to expand it in a Taylor scries vlllid ill some ncighbourhood of the point, This is equivalent to saying that the function is single-valued and possesses derivatives of nil orders at the point in question. In the equations we shall consider the coefficient.. will be llnalytical functions of the indepcndent variable except possibly at ccrtain isolatcd points. An ordinary point It = (I of thc sccond order differcntial equation (2.1 ) y" rx(x)y' P(.-c)y = 0

+

+

is onc at which thc coefficients rx. pare llnalytical functions, It can be shown that fit ally ordinary 1)oint every sol1lfion 0/ the equatioll is allalytic. Furthcrmore il the Taylor e;rpallsiems oj rx(x) allel P(x) tire valid I'll the range I x - a I < R the Taylor expallsioll oj the solutioll is Villid lor the same range. As a conscqucnce, if rx(x) and P(x) arc polynomials in it: the series solution of (2.1) is valid for all vnlucs of x. When, as is llsunlly the case, rx(x) and P(x) arc polynominIs of low degree, the solution is most ensily found by assuming 1I power scrics of the form

•

y = :E cr(x - at

(2.2)

~,

for the solution and determining the coefficients co' c" c~"'" by direct subst.itution of (2.2) into (2.1) lind equating coefficients of successive powers of x to zero. The simplest equlltion of this type is

+

y" y = O. (2.3) Substituting a solution of the type (2.2) with a = 0 into this I) See, for CXlllllplc, Eo L. Ince, Ordimlry Di/lrrtlllia{ E'/rul/iOll.!, (l.OIlSIllIlIlS, 19:!7). Chill. V II; E. Goursnl, A COl
6

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

§3

equation we find that, if the equation is to be satisfied,

•

:E r(r ....0

The series

011

~ I )C•.'I.:'-2

• + ..-0 1: c,X' =

O.

the left is equivlIlent to

•

:i: (r

,-0

+ I)(r + 2)cr+zV'

so that, equating coefficients of a:', we sec that the c
+ l)(r + 2}Cr+2 + c, =

O. (2"') The eocffieienl~<; co' c1 arc determined by the prescribed values of y, y' at x = 0, nnd the others arc determined by equation (2..q. From this relation it follows that the solution is

) (XX)" y=co (1- X X 21 41 -,,, +c 1 x- aJ +5i- .... (_.5)

+

All equation of the kind (2.,~) which determines the subsequcnt eoefficienls in terms of the first two is called n recurrence rclation. 3. RC~1I1nr Sin~t1lnr Points. If either of the fUllctions o:(x), fJ(x) is not lInalytie at the point x = (I, we say that this point is n. sill~l1lar point of the differcntinl equation. When the functions Gt:{a:), (J{a:) arc of such n nature lhnl the differential equation may be written in the form (x - a)2y" (.1l - a)f1(a:)y' q(x)y = 0 (a.l) where 1)(.'1.:) lind q(x) nrc analytic at the point a; = (I, we say that this point is n re~ular sin~lIlar point of the differential equation. If x = tl is n regular singular point of the equation (3.1) it can be shown Ulllt there exists nt lenst one solution of the form • y = 1: c,(:z: _ a)V+' (3.2)

+

+

....

7

INTRODUCTION

whieh is yalid in some neighbourhood of Z = Q. More specifically, if the Taylor expansions for 1'(Z), q(z) nrc yalid for I z - (/ I < R, the solution (3.2) is valid in the smlle range. Putting

•

p(x) = E 1Jr(x -

(Iy,

•

q(x) = E qr(a: -

.-0

(3.3)

(I)"

....

and substituting the expansions (3.2) and (3.3) into (3.1) we see that for the equation (3.1) to be satisfied we must have

•

Ec.bl

'-
+ r)(1' + r-1 •

)(x- tI)f+r

•

....•

....•

.... Ell.(x-a)'. ECr(~+r)(z-a)4'+'

+ .... E q.(a:- a)' Ec,(z- a~r = ....

O.

(3.,~)

Equating to zero the eocrricient of (a: - a)' we have the rcilition co~(e -

so thnt if

1:0

*

1)

+ 'PoCo + qoco =

0

n we hnve the quadratic eqwltioll (3.5)

for the determination of (1. This is known as the indicial e
cree + rHe

+r-

,

1)

..

+ E,f7'.(!? + r -

.~)

+ 'f,}c

H

= (}

which may be written in the form

c.{(e + r)(e+ r - I) + 1'0((1 + r)+ qo}

,

+ I: {P.(e+ '-'1 + q.),,_,~ O.

-,

(3.0)

6

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

§J

Equation (3.5) gi,'es the two possible values ~I' ~! of q. If we lake one of these ,'ahles, !Jl say, :and substitute it in the recurrence relalion (3.0) we obtain the corresponding \'Illue of the coefficients c, and hence the solution ~

YI(Z) =!: c,(z - a)r+t>l. ~.

a similar WilY the root to the solution

Tn

(!~

of the indicial cqualion Icads

~

y~(.r)

....

= }"; c;(:c _ (/)"+9•

Thrce distinct cases arise nccording to the nature of the roots of the indicial equation. el - (!1 'lcither UTO flOT (III illtcger. In these circumstances the solulions Yl(X) and y:(x) arc linearly indepcndent and thc genernl solution of equn.· tion (3.1) is of the form ClI4e (i)

•

y = }"; c,(a: - aY1"~1 ,..(1

ClI4e (ii)

(ll

=

+,_0 }";• c;(x -

a)~.

(3.7)

(!2'

If (!\ = (]2 the solutions Yl(:t) lind Y2(.r) arc idcntical (except, possibly, for n multiplicative constnrlt). The gCllernl solution of the cclulLtioll CUll bc show.. to be yd:t') Y1(X) where

+

Case (iii) (!2 = {JI - 11 where II is (I positive i"tegeT. In this easc nil the coefficicnts in onc of the solutions

INTRODUCTION

9

from some point onwards nrc either infinite or indeterminate. It can be shown thnt the appropriate solutions nrc

Yd x ) =

(x -

a~1 ~ er(x -

nY.

r_O y~(x) = g"Yl(X) log (x - n)

-

+ (x -

I

~

a)fo

(3.9)

....

E ur(x-ay.

where g" is the coefficient of x" in the expansion of

+ a )} ~ cxp

X"H

{YI (.1:

[f' ] IIp(tt)dn. 0

M

It may happen thnt g.. = 0 in whieh cnse y:(x) docs not eontnin 11 logarithmic term.

4. The Point at In[illity. In mallY problems we wish to find solutions of differential CCluations of the type (3.1) which nrc v"lid fur large "llllles of x. We seck solutions in

the form of infinite series with variable....!.... If we make x the transformation 1

x=1"" the 'point at infinity' is taken into the origin 011 t.he ~-nxis. With this change of vnriable equation (3.1) becomes

lP.Y+{2 -~I1(2.)Jd!l+.!..p(2.)y=o dO'

,

0'

,

d,

"

,

(.u)

~;-I _ ~~(~-l). ~P(~l) nrc both OCt) as ~-+-O and analytic in ~ then ~ = 0 is nn ordinary point of equation (.)..1) and we SHy thnt x = CO is nil ordinnry point of equation (2.1). Hetllrllillg to the original independent vnrinble we sec tllnt the eomlition for the point lIt infillity to be an ordinary point of C
If

o

«(x)

=..:. + O(r-'). x

fJ(x)

=

OCr) as x-+- CO

(4.2)

10 THE SPECIAL FUNCTiONS OF PHYSICS AND CHEMISTRY

§5

The corresponding solutions nrc of the form

Similarly if,

liS

m -+ 00 (,'.3)

where 0: 0 _ flo firc constnnts wc sny that the point fit infinity is a regulnr singulnr point of the equlltion (2.1). The corresponding indicial cqulltion is ~'+ (l -

")~

If nle roots of this cqunLion nrc

+ p ~ 0.

lh the solul:ions of (2.1) valid for lnrge vfllues of x nrc of the form (II'

•

•

•-0

.-0

( ) = '" ' ' 'erx ' - . - ,. ).. errq-r (» = .... yl:r I , V:. x

(H)

5.

The Ilamma function and related functions. in developing series solutions of differential equations and in other formal calculations it is oftcn eOll\"enient to makc lise of properties of gammll nllel bcta funcLions. Thc integral

r(11)=f:e-~x'l-\{l;t'

(5.1)

converges if /I > 0 and defines the gamma fUllction. Similarly is III > 0, 11 > 0 the ucla function is defined by the equation ll(m, 11) = J:X"l-I(l-X)"-ldx

(5.2)

It is then easily shown that l)

(i)

(ii)

r(l) T(11

~

1

+ 1) =

1Zr(ll)

I) For proors or these results the render i~ rderrcd to n. P. Gillcspie, lllf.gratial/, (OHver IllLti Boytil, IDSI, Pfl. 00-U5.

INTRODUCTION

+ 1) =

11

(iii)

F(n

(iv)

B{m, 1/) = :! J~" sin 2JR - 1 0 COS 2 "-l 0 tiD,

Iv) (vi) (vii) (viii)

(ix)

DI.

)_

III, It

rm~

-

III if II is

ft

positive integer,

I'(m)r(II) '/l)'

+

F(m

v'n,

F(],)T(l -1') = : t cosec (r'Tr), 0 < p < I, r(!)r(2J1)=22~-lr(Il)F(II+!)- the duplication formula,

1'( :: +1) = I'lin

"~.

"I,,'

(z+ 1 )(z+") ...

Iz+,,)

I;:>0.I

When /I is 11 ncgntivc fraction F(II) is defined by means of equation (ii); for example

By mellllS of the result (ix) we can derive all ill\.crcsting expression for I!:ulcr's constnnt, y, which is defined by the equation y = lim (1 n_oo

+ ~ + ... + -.:. -

log 'II) = 0.5772

(5.3)

/I

From (ix) we have

~{lo~r(=+ l)}= '"

liz

lim (100'11- - ' - _ - ' - - ... _ -'-) .::+1 z+2 :::+11

n ...., . , . .

so that letting z --'10- 0 we obtain the result (.iA)

lLnd from (5.1) we find

y= -

f:

e- l logfdl.

(5.5)

12

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

§S

Integrating by parts we see that

f ,oc

f

log tllt = log::

+ for. -dl , t

so that - Y = lim

.~

(f• - t

""/:-I

dt

+ log z . )

(5.6)

Closely related to the glllumn fUllction are the cxpOllcntial-integrnl ei(x) defined by the equation ei(x) =

f

O'-" - d.. '"

(x> 0),

(5,7)

II·

and thc lognrillllllic-integrlli li(x) dcfined by li(X)=f'l

dU

o og

•

(5.8)

I~

Si(.r) Ci(x)

-1'-

-1

- 2 "--=~,,--,--~:-::c'-:-----:-~~----' Fig. t Variation of Ci(~1 nnd Si(z) with ~,

which are themselvcs connected by the relation ci(z) = - li(c"').

(5.9)

Other integrnls of importance arc the sine and cosine integ-nils Ci(x), Si(r), whieh arc uefincd by thc equations

INTRODUCTION

§ 5 ,a)

Ci{.v) = -

cos

Il

Si(x) =

J, --du, "

13

"sin I S,--,,, " U

(5.10)

and whose variation with J: is shown in Fig. 1. In heat. conduction problems solutions CUll often be expressed in terms of the error-function

" S'0 c-~Idrl,

crf(x) = .~

vn

whose vnrinl:ioll with

It

(5.11)

is exhibited grnphically in Fig. 2'·

'-0 ,--~-~--===-----, 0·8

+0-.

iI

(H

0·2

o

3·0

I·(l -T_

Fig. 2

Vurintion of erf(z) with x.

Sirnilndy in problems of wave Illotion the F,'csncl intc~rnls

• A. C. Aitken, SlalistiMI :'f(J/J,tntatiu, (Olivet.f1. Royd, Seventh Edition, In;,:!) p. G:! gi\'ClI n shott tallie of \'nluClI of erf (at).

14

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

C(x)

=

s:

cos

(1 1lU2 )dll,

8(x)

=

J:

sin (!nIl 2)du

§5

(5.12)

occur. The \"llriatioll of thesc functions with x is shown in Fig. 3. 0·8 r----r--~-~--~-__,

--C(x) S(,T)

0·0

o·,~

U·2

II

Fig. II

:l _.1'_

.,

VariaUon of the Fresnel integmls, C(x) and S(x}, with x.

The importallcc of these fUllctions lies in the fuet thot it is OftCII possible to express solut.ions of physical problcms in terms of thcm. Thc corrcsponding Ilumerical values can thcn bc obtained from works such ns E. Jahnke !lnd F. Emdc, 'FlIlIkt;ollclj{a/dll' ('I'cubllCI', ].eipzig, 1H:J3) in which they IIfC tabulated. EX.~:UPLES

I.

Show that, in Iillhericnl polrLr coordinates r, 0, rp defined by x _ r ~ill 0 eo, rp, y _ r sin 0 Sill If" : _ r cos 0,

LJlllbce's equutiOll become'

INTRODUCTION

(r' 'lIrloP) + ~illO _'_ ~ (Sin 0 i'l~') -I- _1_ i'l''f' _ 0 i'lr i'lO i'lO sin'O op' ' ~

"

and pro\'e that. it possessCli soilltioll~ (If the form f·t''""B(cos 0), where el}l) !llIlisfil:s the ordinary ,Iiffcrtlliial equation (I - II') d'e __ - 2)1 -fie

<1}J

1I}1'

2.

+{

11(.1

+ I)

-

sinh

~

- m' --

I - },'

1

fj _ O.

Show that if II

.:l: _

cosh'; cos 'I, Y _

Laplace's e'lilalioll

i'l'V' _ <:IP

lISSUIllCS

+ O'VI _ <:11/'

(I

sin

1/, :: _

::

thc form

+ 1I'(('osh ' '; _

1 " ,'1 .. cos' ./) <:_ 0: '

o.

Deduce thnt it hRS solutions of the form !(i;l!l'l)rl" satisfies the c(lunlion

d'/ _ +

whcre 1(1/)

(G+ IUq cos '2,/)/ _ 0

d,/'

in which G is 3.

1I

{''I''l:!'.!.

(.'unstant of sepllrntion nnd q _ -

Purnbolic (.'()flrdinnlc!I .;, '1> 'P nre ddined by ;l -

,/(';1/) cos tp. !I .. ,/(';IIl sin rr, .:: -

'I>.

H; -

Show thllt ill these coordinllles l..llphlce's efillation becomes .,

.; + 1/

,(,,")

<:1.;-

."

(")

I 0''1'

~ 0'; +.; + '/ <:II/ '/ OIl + .;,/ ~;p.

-

0,

Prove that if /,'.("') is II solution of 1I1e efillation d'f' dP ._+_+

d.x;'

(

m') fo'-O

11 _ _

dx

4:1'

then p.(';)/"_.{r/)e±I"'9' is :1 wlutioll of Lnl'lnce'!1 c'l'",Uon, -I,

Dcfinin~ (''OS,1', ~ill

x to he lhe solutions of fill)

---:.. + y

_ 0

d;c'

which res]leclil'cly lIrc I, 0 when

:I' _

n, provc

16

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY (i) oos( - ~) _ eo! z, sin( - zl _ - sin z; (ii) oos(z + r) _ oos;r oos z' - lin;r sin r; :1:') _ lin z rosz' 005% sin r; (iii) sin(z (i\") cos'z + sin1z _ I;

+

(,,)

+

!!.. (sill.1:) _

'"

5.

!!..- (COI.1:) _

co, z,

Sill Z.

-

'"

The 0111)' sillj:tulnrilics of Ihe differential equation

Y"

+ IJ(z)y' + 'J(z)y -

0

lire I't'gulllr singularitie!lllt:r _ I of exponenlll 11, a.' nml nt;'l _

ofe"IKlllelL!:.s p, lhllt p _ - 11:, (Zl -

- 1

p' tile point lit infinity Ixoin.llllil ordinary point. Pro,'c P' __ 11' lUlIl lh:lt the diffel't'nli:l.l efllmtioll b

I )oy"

+ :!(.l: -

I I(z - a. - 11')y'

+ ·w'y _

0

Show thaI the IOlutlon I, y -',

(z - ')" \.i"+l

- ')"' + c, (z ~

"'here c, nnd c. are eonst(lIltll.

6.

Appl)' the method of solution in series to the equntion

ztPy -

+ (a

-z) dy __ y _ 0 dz' '" ihowins tlmt, nenr z _ 0, y _ Au + nil whcre u is a MlIcinurin seriCll and v _ z'-·e-. In i. lIot lin integcr). 7.

Find two 10lutillliM of the eqlllllloll (%1

liy + 2.1') _ll'y + __

lUi

IU

ill the foml y _

k(k

+ l)y _

0

.-.• a.z"·' ~

Sho.... that, if k 15 II f"O!Iit;"e intC'j.!cr. one of Olese IOI"tioDt is IKllynominl I -I- I.. (k

+

I)~I (k .. _I

(k

+n _ "+

I)! (~)A 1)! (:!n)!

tJll~

INTRODUCTION Pro\'l~

8.

"

that if, i! lUI integer 111111 a ii fmcLiollnl r(o _ .)_( -1)' l1tl)r(1 - a). r(1 u +.)

9.

Show thnt (i)

(ii)

(a - 1)(:1)._. _ (or: - I). (:I).

(<<).-. - ( - l

r "(Ic-'",,-,-= or: II),

II!

(iii I - - - _ (- 1)'(- 'I). (II -.)1

where (a),_ «(<<

10.

+

1)(.%

+ :!) ... (or: + r

-

I)

I'rove lhnt

ei(x) _ - y - toga: nnd dedul..'e llltlt CJ(.r) _ i'

Si{x) _z

+ log a:

...

+x

.'.

- -

2.21

lI:' __ 2.21

...

+_

.l..l!

-_+ __ ... n.:u

5.S!

;tI + __ ...

3.:11

- ...

Cll,\I'l'Ell II

HYPERGEOMETRIC FUNCTIONS 6.

The 1

HYfler~eoll1etrlc Series.

The scrics

+ """"t' •. PY' I: + .(.1 + I)P(P + 1) • + . ~y(y + 1) ;rv

• --,

(G.l )

is of great importance in rnnthcmatics. Sincc it is an obvious generalisation of the gcomctric series 1

+x+tt2 + ....

it is called the hyper~cometric serics. It is readily shown that, provided y is not zcro or 11 llcgati\·c intcgcr the scries is nbsolulcly convcrgent if I x I < 1, divergent if I x! > 1, whilc if I x I = 1 thc series con\'crgcs IIUsolutely if y > ex p.• ) It is convcrgcnt when ,'I: = - I, providcd that y > ex (J - 1. H wc introduce the nolnl ion r(ex r) (ex). = 0:(0: 1) _. _ (0: r - I) = -r~ (6.2)

+

+

+

+

+

we Illny wrile Ihe series (0.1) in thc fOl'm x) ~ " (·),(PI, x' (6.3) ... 1(.)" ,.-or.y. tlie suffixes 2 and I denoting thnt thcrc lIrc t\\"o parameters of the typc IX and onc of t.hc typc y. We shall gCllcrnlise this concept at n Intcr stnge (§ 12 below) but it is advisable at this stnge to dcnot.e t.he 'ordinary' hypergcomet.ric function by thc symbol 2"\ instend of simply F, if wc nrc ~

,.'I (., p." y'

.) See J. !II. 11)'slop. JlI/illilt Suit!t, FiUh Edition, (Oli,-cr k lloyd, IflM) p. 50.

"

i

HYPERGEOMETRIC FUNCTIONS

6

19

to nvoid confusion Inter. From the definition (0.3) it is obviolls that

A significant properly of the hypcrgcolllciric series follows imlllediat.ely from the definition ((UI). We have It

(Ix ~

P( 'I Cl':.

Now (<<)t+1 = !X(IX

+ l)r

so the right hund side of t.he Inst equlltion becomes

.p i; (.+ l),(P+ I),., i'r-o

r!(y

+ 1).

showing l.hnt

d, I:I.{J dx 2!'\(CJ.,Pi y; a:) = Y2Fdo:+1,P+l;y+l;IV).

(0.5)

It should nlsa be observed Lbnt (n.a)

so thnt

,d !f'\(a., [ IX

Pi r i xl]

"'~O

=

~{J. i'

(G.7)

Sc\'crnl well-known clclIlcnl.nry fUlIctions CIlI} be expressed as hypcrgcomctric serics; examples of them nrc given in ex. 1 below. It sllould be noted that, if we adopt u certain convention, l\ hypcrgcomct.ric series can slop nnd start lignin after 1\ lIumber of zero terms. For example, consider the hypcrgeometric series 2.FI(-llj Uj -11-11I;.7.) wherc bolh 1/1 lind

20

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

§7

/I. nrc positive integers and b is neither zero nor a negative integer. Because of the oceurrellee of (-'1/.). in the numerator in the expansion in powers of x it is obvious that the (11 1 ).th. term of the expnllsion will he zero, ami we arc tempted to think that cvcry subsequent term is also zero. Tf we note that, as a result of ex. 9(iii) of Chapter I,

+

(-'Il) nl , ) _ (+ I' (n+m-r)('Il+m-l'-l) ... (1/.-r+ 1) 11 m,. 111/1.. (G.8)

when the form on thc left is not of type 0/0, and if, further, we assume that it still hm. tile value on the left when it is indeterminate. we see that we may write 21;\(-11, h; -n-mjx)

') ' ) ... ( 1- - ' ) -1-' (b)," (6.1l) = "? 1. (1 - -(1- ::O-::C, .... 0 11+711 11+11I 1 11+1 r so that although the series stops at the ll-th. term it starts up aguin at the (1I+"'+1)·th. term. For instance, l)

~F,(-2 l ' -5, x)~ 1+':-.+ v

" .

5

1

]

<,

10

10

:1

-x"- -,x3 _~1_x8+ ...

7. An Integral Formula for the Hyper~eometrlc Series. In order to deri\'c some further properlic.<; of the hypergeollletrie series we shull first of all estnblish lin expression for the series in the form of nn integrnl. It is readily shown that

(fJ). = B(f1+r, y-(1) = 1 fl (1 _ t,)y-P-11.P+H rll. (y), B(P, y Pl IJ(P, y-P) 0 from which it folloll's that.

~

((J:!. X'JI (1-J)t-,8-lIP+.-Idt.

(1).-01'.

0

Interchanging t.he order in which the opernl.ions of sum·

p

HYPfRGEOMETRIC FUNCTIONS

21

mation and integration are performed we see that 2/i\(&:, Pi Yj x)

="-"-,,, fl (l~t)l'-IHt~-1 { IJ(P, y-Pl 0

~ (tt), (;l:t)T

.-0

>'!

I

(It.

Using the fnet that

~ (IX), (xW = (I _ .1"1)-". ,.!

,-0

we have the integrnl formula

2""(1X, pj ri x)

f'

(J) o(l-t)y-j1-I/"-I(J-3:I)-.1 dt • (7.1)

I = lJ(P,y

valid if I x I < I, i' > {J > O. The results hotd if z is com· plex pro\'idcc1 L1mt we choose the brnnch of (I - xl)-:O in such /I WilY that (1 - .r/)- --J> I us t -)- 0 and !l1(y) > !l1(P) > o. The flrslllpplicnlioll of (7.1) is the dcrivlllion of the vallie of the hypcrgcomctric series with unit argument. Pulling It = 1 in(i.1) we have I tFt(a,{Jii'i 1 )= H(P,y

~

f'

fJ) oP-t)l-2.-"-lt"-ldt

B(jJ, y-.-p) H(P, y P)

Cl P > 0, (J> O. If we express the bela function in terms of gamma fUllctions we have Cnuss' Theorem

if}' -

,"';(., p; !'ow if

IX

_ rly) I'ly - • - Pl

y; 1 ) -

Fiy

.j I'(y

Pf

= - ". n negative integer, we have

r(y - . - Pl r(y P) ~

('I-Pl.,

(1.2)

22

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

§7

so that equation (7.2) reduees to t

P1 (-

R. •• ". fI'

) _

i"

I

-

(y - Pl. (y) ..

whieh is known, in clemen wry mathematics. as Vandermonde's theorem. Again. if we put x = - 1 Ilnd IX = 1 {1- 'Y we have, from equation (i.l)

+

R. R _ 2F I(IX. 1'1/' a.

+1,._1)_ T(I + /1- 0) J'I _ - F({1)l'(1 a.) 0 I

').' ,., / t d.

/

=

If we write ~ /2 in this int.egrnl we see that its vnlue is tB(~P, 1 - a.). Using this rcsult lind the relation !r(!p)fF(fJ) = F(I tfJ)jI'(1 Jl) we huve Kummer's theorem

+

+

T(I +p-o)r(l+ IPI

,F,(o,P,P-O+I;-l)~r(l+p)/'(I+!P

or

.

(7.31

Further we can deduce from the formula (i.l) relations between hypergcometric serics of nrt:l.Illlent x llnd those of argument :rJ(z - I). Pulling T t in C(luation (7.1). and noting that

= ]-

{l- x(1 - T)}-
we sec that

whence we have the relation

{I - "-I -"-T}-'

I,

HYPERGEOMETRIC fUNCTIONS

23

find, by symmetry, the relation

Pi "i IV) =

2FdIX,

{1-:r:)-P21·\ (r-IX,

pj )'i a:

:1:

1)'

Using the symmetry rellltioll (GAl nnd equution with a:: rcp1:Jccd by ;rJ(a: - I) we sec that 2P l(a:,y-{1;Yi x

,1:

1)=2 P l(r-P,Ct;Yi . t: .v

(1.·~)

1)

p, y -

= (I - .x)J-1I 2P dy -

(1.5)

Ct; )/; x).

so that 2 P dIX,

fl; y; x) = (1-a:)1'-I>:-1l 2 /i\(y-iX, y-p; Yi .'c). (7.0)

Tf we put x =

t

2P l(a:,

in equation (TA) we obtain the relation

P; Yi t) =

2" 2P I(c:r., y-fJ;

)Ii

-1).

'fhe series all the right hand side of this cquntion cun be clcri\'cd frolll equation (7.3) provided either that

y = y - {J -

r.t

+ 1,

Le.

fJ =

1-

iX,

or that y

~

_ -

(y -

P) + 1,

i.o. y

~

i(- + P+ 1).

We then obtain the formulne (1.1) (1.8)

8. The Hypcrgcomclric Equation. Tn certain problems it is possible to reduce the solution to lhnl of solving the second order linear differential equation rr-y tt(l- x) d:r:~

in which

0:,

+ {y -

(1

dy + a. + ,8l·'!:} dx -

a.,8y = O.

(8.1)

,8 nnd y nrc constnnts. For instancc, thc Schro-

24

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § a

dinger equation for n symmetricnl-top molecule, whieh is of importnnce in the theory of molecular spectra,l) CUll, by simple transformations, be reduced to this typc. An cquation of this typc also !Irises in the study of the flow of compressible fluids. In nddit.iOll certain other differential equations (such IlS that occurring in ex. 1 of Chapl.er 1) which nrisc in the solution of boundary "nlllc problems ill mnt.hemnieal physics can, by It simple change of "ariable, be transformed to an equal.ioll of type (8.1). Indeed it enn be shown that any ordinary linear differential equat.ion of the second order whose only singular points arc regular singular poinL", onc of which may be the point at infinity, ellll be transformed to the form (8.1). For that reasoll it is desirable to investigale the nature of t.he solutions of cquntion, which is called the hyperQ,eoll1ctric cquation. We may write the hypergeometric efjuation in the form it~lI"

+ x(l + + It~+ .. .){y -

+ +

(ex {J 1 )x}y' -«{J:c(1 +a:+x~+ ... )y= 0, so that, in the notation of § a, we see that ncar x = 0 it

Po = y, '10 = 0, nnd the indieial equation is

(22+(y-l)(?=0

with roots (! = 0 and (? = 1 - y. Similarly, the equation CUll be put ill the form (.T-l

)'y"-(x-l)(y-.-P-l-(y+.+P+l )(.T-l >+ .. .)y'

+.P(x-l)(l-(x-I)+ .. .}y~O, with indieial cquation

e'+ (.+ P- yl"

~

0,

of whieh the roots arc (2 = 0, Q = y - « - (J. Finally in the notnlion of § .~ wc hnvc for large "lillie:." of x I) See, for eXlllllple, I.. PlIuling lind E. n. \\"il!KIn. Illlr(J{/ucIiOli 10 QUllfltllm 1l1fChcllliCJ, with AIIlllka/iolls 10 Chemislry, plcGr:\w-Ui11, New York, lOa,,), JlJl. 275-2~lU, (Iud ex. 10 ucla\\'.

§8

HYPERGEOMETRIC FUNCTIONS

a(x)~ (.-I-~-I-

2S

P(x)~"!,

1>,

find so the indieinl equation llppropriate 10 the point at infinity is e' - (a -I- P)q -I- ap ~ 0, with roots I'J., p. Thus the regular singular poinL~ of the hypergeometrie equation are:(i) x = 0 with exponenL,> 0, 1 - y. (ii) x = co with expollelll.s (I., p. (iii) .'1: = I with exponent.s 0, y - (.( - p. These facts ure exhibited syrnholielllly by denoting the most general solution of the hypCI'gcollldrie eq\lntioll by u scheme of the form

y=r!g

1-y

~P

~

(8.2)

y-.-p

The symbol on the right is called the Ricmallll-P-fIlIlCtion of the equation. We shall now consider the form of the solutiolls in the neighbourhood of the regular singlllnr points. (a) x = 0: Corresponding to the root (} = 0 we have a solution of the form

• y=:i.:c,x'. ,..,

Substituting this series into eqllHtion (8.1) we obtnin the relation

•

(l-x)1:c,r(r-l)x H

,-0

• • -I- {y- (a-l-p-I- I ),t'} 1: r,I',I,r- 1 - rJ.fJ 1: crx' =

r_U which is readily seen to he equivlIlenl 1.0

•

,_u

:£ {cr+l[r(r+ 1 )+(1'+ 1 )yJ - c,(r+rJ.)(I'+fJJ },t" = ~O

0,

U

26

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

so that (r Cr+l = (r

+ a)(r..L.. P) + 1)(r T 1') cr'

§8

(8.3)

from which it follows that -

(al,(p)"

',- (I' I, r.I

(8 .•')

o·

It follows that thc solution which reduces to unity when x= 0 is y

~

1

+ yI!x aP + a(. + J IP(P + J) , + 1'(;,+1)21 x ... y

i.e.

2Jo'I(IX,

=

Pi

Yi x).

(8.5)

Similarly, if 1 - I' is not zero nor:l. positive nor negativc integcl', thc solution corresponuing to tllc !'Oot IJ = I - Y is y =

•

~

c,m1-1'+'

~"

where

•

(I - .'1:) L c,(r+ 1- y)(r - y)x'-Y

,-0

• + {y- (et.+,B + 1 ).c} ,..0 ~ c,(r+ 1 -

+ •

y):tr-y -et.{J L c,xl-y+' = 0, ,-0

which is equivalent. to

•

E,,{('+ 1 - YH'-Y)+Y('+ I-yllx'''''-

.-0

•

_ L c,{(r+l-y)(r-y)+{et.+P+ 1 )(r+ l-y)+IX{J}x'-Y+!=O, ~O

implying that Cr+l=

(, + a - Y + 1)(' + (r+l)(r+2

Py)

y + 1)

c,.

Comparing this relation with (8.3) and and taking Co = 1

27

HYPERGEOMETRIC FUNCTIONS

§8

we sec that this solution is xl-YzF1{a.-y+l, {J-i'+'!j 2-y;

xl·

Combining equations (8.5) and (8.6) we sec that the general solution valid in the neighbourhood of the origin is

y=A ZF\(a.,

Pi y; x)+B:rI-Y zlr't(a.-,+l. j1-y+ 1; 2~y; xl, (8.71

provided that 1 - Y is not zero or n positive integer. If y = I, the solutions (8.5) and (8.6) arc identical. If we write

und put YA:c) = Yl(X) log x

+,-,1:• crx.

we find on substituting in (8.1), with y = I, tlillt

1,+1)',

.-+1

-'I

a.

+P+ 1)'r + I·I,IPI,I·P-·-P-'I rl{r+ 1)1

0

from which the cocfficicnL<; c, may be determined. A similar procedure holds when 1 - Y is a pos[t[\·c integer. (b)x = 1: If we let

dy

~

= 1-

x, equation (8.1) reduces La

dy

III -II ,II +,(tt+P-y+l-ltt+P+1)} <11- ttPy ~

0

which is identical with equation (8.1) with y replaced by IX (J - y 1, und x by $ = 1 - ,'t. Hence it follows from equation (8.7) that the required solution is

+

+

y=A ~J;\(a,{J;a+{J-i'+l;1-x) + H(l-x),.....-
28

THE SPECIAL fUNCTIONS Of PHYSICS AND CHEMISTRY

§9

which gives

..

• ,

(1- x) L c,(r+a)(r+ a+ 1 )X-r-Cl-l

,..•

•

....

- {y- (a+p+ I )X} 1: (r+a)c,X'"""I"-
•

•

r-O

,_0

I: c,(r+a)(r+l1-y+ 1 ),rr-"-l = L c,r(r+a_p).rr-Cl,

whence it follows thnt

(r+a)(r+a-y+l) p+l)c"

cr+t=(r+l)(r+a

which

III

turn is equivnlent to

(a),(a - i'

+ 1 ),

c,= r.a'I P+) 1, ',. Tnking

Co

= 1 we obtnin the solulion x-":P1(a. a- y

+ 1;

1

a-p+ 1; -). From the symmetry we sec thnt the other x solution is x-fl2Pt(P, P - y 1; P- a 1; :), so thnt

+

+

the required solut.ion is

y = Ax-' 2P I (a, a - Y + 1; a-p + 1; :) (8.0)

9.

Linear Relations between the Solutions of the Equation. The series in the solution (8.1) nrc convergent if 1.1: I < I, Le. in the intervnl (-1, 1) wherelIs those in the soh Ilion (8.8) nrc convergent ill (0,2). There i.~ therefore nn inlen'lIl, namely (0, 1), in which all foUl' series converge, lIlId since only two solutions of the differentinl eqllntion nrc linearly indepen{lent it follows tlmt there must be a lineur relution valid if 0 < x < 1, between solutions of type (8.7) und those of type (8.8). Hyper~eomelrlc

HYPERGEOMETRIC FUNCTIONS

§9

29

Let 2F,(a,

Pi Yi .1:) =

A 21'\((,(,

Pi r.t+p-y+l; 1-.1:)

+ B(l-x)Y-
Pi a+,B-y+ 1 i 1) -I-

B 2F,(y-cr.:, Y-pj Y-IX-fJ+ 1; 1 l,

and put.t ing x = 1 we have 2/,'I(ex, pj y; 1) = A, if we assume that I > Y> 0 p. (0.1 ) Substituting for the series with unit nrgumcnl from equa· tion (7.2) we sec that A _ F(y)F(y - 0 - P) - T(y o)F(y P)' and that

+

1- _I T(o+P-y+1)T(1-y) - . F(P y+ 1 )F(o y+ 1)

+n

r(y-o-p+1 )1'(1-y) 1'(1 P)r(! 0) ,

so that

n _ T(y)F(o+ P- y) 1'(o)F(P) , whence we find that 2 PI

(0:,

T(y)F(y-o-P) ex)r(y fJ) 2 F I (rx,

p; y; x) - rey

p; IX +fJ -

y+ 1 :

1

-;1:)

F (y-O( ,,_R. y-lX-R+l' 1-.2:) + r(i')r(o:+{J-Y)(I_X)l'_"_~ r(o)F(P) , . ' , ,-, ,-, ,

(0.2) provided that the condition (!U) is satisfied and If we replace

it

by

2.-x ill

O<x< 1.

equation (9.2) we have

30

THE SPECIAL FUNCTIONS Of PHYSICS AND CHEMISTRY

t

9

1 r(y)r(y-x-P) , 1 ,F\(a..{J;y;-;)- r(y a.)r(y (J),l'da..{J;a.+{J-y+l; 1--;)

+ r(y)r(x+P-y) (I-:")~'.I"I(Y-X y_p. y-x-P+ I· 1-:") qX)rlP) X " 'x ' nnd from equntion (7A) ,F1 (<<. (J; y; 1- :.) =

r'~Fl(ll. i'-{J; (;

l-x),

so thnt I r(y)r(y-x-PI F(x, p; y; ... ) = Fly x)Fly Plx" F(x, x-y+ I; x+P-y+ I; I-X)

rr1(X-I )'1'-<'-6 P(y-IX 1-«' y-x-P+ 1'1 -x) + r(y)F(<<+{J-y) r(x)r(Pl '" , (0.3) where 1 < z< 2 nnd I>,> (J. These relations nrc typical of n larger number which exist between the solutions of the hypergeometric C
«+

the equation trnnsforms to one of Lhc Sfime type (but, of course, with differcnt parnlllclcrs). The cquJlt.ioll (8.1) therefore hns twelve solutions of thc types (8.5) lind (8.0) - two for eneh independent vllrinble - each eOll\'ergent within the unit circle. Any olle of these enn be expressed in tenns of ty,·o fundamental solutions. In addition twelvc more solutions of the kinds

F 1(Y-:l. y-{J; y; x), xl-I'( l_z)1'-2-11 ,f.\ (I-IX. 1-{J; 2-i'; x)

(1_X)7- 2 -1I,

can be deri,·ed. The relations between these twenty-fouf solutions of the hypergeometrie equntion nrc of the types (9.2) nnd (9.3); for fI full discussion of them the render is

110

31

HYPERGEOMETRIC FUNCTIONS

referred to T. )1. MncRobcrt, FunctJ'OIl, 0/ a COIII],kt l'ariabI~. p[ucmillnn, 2nd edition} pp. 208·301. 10. Relations of Contiguity. Ccrlnin simple relations exist between hypcrgeomctric functions whose parameters differ by ± 1. For cx:nml'!c if the pnrnmctcrs 0: and P remain fixed and y is ,"uricd we can prO\"c that

p; y; x) + (y-.)(y-P),v,Io',(., p; ,+'; xl - y{y-l)( I-X)2F,(o:, p; y-l j tt) =

, (y-l-(2y-.-P)x),F,(.,

O.

(l 0.1 )

The proof follows from the definition (0.3) for the coefficient of x" in the cxpnllsion of the function 011 Ihe leJt of (10.1 ) is ( _l)(·I.(P)._ (2 _._P) (.I-o(Pl._, y)' (y)"lll Y (Y).-t(II 1)1

+(

)(

y-a; y-

+

P)

(x)_, (P)-,

b+ 1 ).._t(1I

( I ) (·).(P).

t)! -YY-

(,

1),,11!

(-1) (·).-,(P). , y Y (y l)Il_I{n I)!

and it is not difficult to show that this is zero. In nnolhcr kind P nnd y nrc kcpt constant ami vnricd. One such is {y-.-p- (P-.)(I-x)) ,1",(" p, y;

+ ct(I-X)2FI(:X+ I, Pi ri xl -

IX

is

xl

(y-tX):.l"I(tX-l,

Pi ri

x) = 0 (10.2)

the proof of which is similarly direct.

In the thjrd type of relation y is kept constant and Cl and fJ \'nry. Dne of the simplest among these relations is (a..-fJ):.F 1 (ct,

P; ri x) =

IX :.Ji\ (<<+ I,

Pi ri x)

- P,1",(" P+'; y; x)

(10.3)

The proof of these relations is left to the render; further examples nre gh'cll below. (cxs. 3, 4)

32

THE SPECIA.L FUNCTIONS OF PHYSICS AND CHEMISTRY

t

11

J 1. The ConrJuent Hypergeometric Function. If we replaee;z; by zl{l in C(I'"ltion (8.1) we see that the hypergeometric function

,10',(_,

P,

y;

xlP)

is a solution of the differential equation

X(l- ;):~~+{Y-(I +cttl)x}~~ -

cty=O

so t1U1t lelling {l-+ 00 we see thnt the function lim :1"1(ct, (l; Yi

xlP)

(II.I)

If-H1I>

is

8

solution of the diffcrentinl equation

x

,r-y

dz2

fly + (Y-X)tlx -cty= O.

From the definition of

({l)~

lim

(11.2)

we see thnt

(~~ =

1

If-· ... t'

so Ihnt the fUlletion (11.1) is the series ~ (Il), . x~ ~_o

(,,), rl

(11·:11

nnd this sc.-if's we denote b~' thc symbol 1 PI(a; y: 11:). This fUllction is cullcd II confluent hypcq~eol11etrlc rUIlC tlon, and thc CCjlllllioll (11.2) is thc confluent hyJlergeometric equlltion. Equations of the type (11.2) occur in 1l11lthematical physies in the discussion of boundnry 'falue problems in potential theory, Illld in the theory of atomic collisions (see examples 13, 1·' below). It is readily nrified thnt the point x = 0 is a regular point of the dirrerentiul eqllutioll (11.2) ancl that. in the nolfitiou of § a, Po =)' lind 90 = O. The indiciul equlltion is therefore M

33

HYPERGEOMETRIC FUNCTIONS

§11

ele + y -

I) ~ 0

with rools 0 = 0 and (! = 1 - y. Corrcspol1~ling t.o the root e = 0 there is a solution of the form

•

~

VI =

C.;l;r j

,-0

substituting this solution in equation (11.2) and equaling to :l.ero the coefficient of x· we [illd that Cr+1 = (y

Putting Co

=

(IX + r)e. + r)(r -I- 1)"

1 we sec that

,,

(rx), (y),

1

~-.-,

r!

and if y is neither zero nor n Ilcgati,'c integer the solution is YI{X) = tfo\(o:;

j'j

x).

(llA)

Similarly, the root f! = ] - y, lends, if 1 - i' is neither zero 1101' II positive integer to a solution of the type y~(x)

rr

= x 1-

•

1: c,z'.

y

~o

lI'e write yz(xl =

lind substitute in equation (Fu

x-I" (X·

(11.~)

xl-"l

u(x),

we find that I/(x) satisfies the

lilt + (2 -y-x)-I x (

(a:- r+ l)u = 0,

which is the same liS equation (11.2) wit.h y I'cplnccd by 2 - Y and IX replaced by a: - y 1. We know from

+

equntion (II ..~) that the solution of this equation whieh has 1 j ~ - Yj x) value unity when it = 0 is It = I Fl(a: - y so thnt

+

(11.5 )

34

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTR Y § 11

Thlls if Y is neither 0 nor an integer the gencral solution of Clluation (11.2) is y(z) =A I P1(lX; r; .:z:)+ Bxl-y I F1(lX-Y+ 1 i 2-Yi z}, (11.6) whcre A and B are nrbitrnrr constants. In the exceptional case Y = I we have YI(.:z:)

=

IPt!lXi 1; x)

(11.7)

obtnined simply by pulling y = I in equation (11 ..q. For the second solution we writc Y~(''l:) = YI(X)

log It

+,.,~• c~xr.

(11.8 )

Substituting this expression in equation (ll.~) nlld pUIting we find that the unknown coefficients c~ must be such thnt

Inserting Ule "nlue of y(x) from equation (11.7) we see that these coefficients arc determined by the rceurrenee relation cl =l-IX,

(r+l)~C~+I-rCr=(l-IX)r!(;~rl)1

(lUI)

The complete solution is therefore given by y= AYI(ro)+ By~(x) where A and 11 arc fil'bitrnry constants lind the functions Yl(.:z:). Y2(.l') arc defincd by equations (11.7), (11.8) and (lLfl). The complete solution when y is an

integer nUlY he found by a similar method. If in equation (11.2) we pill y(.:z:) = ,r1"t'lzlI'(.:r)

(11.10)

we find that the function lI'(x) satisfies the diffcrential equation

tr-W dx' +

I-t+.-+ k !---..---

m~}

lV(x)~O.

(11.11)

HYPERGEOMETRIC FUNCTIONS

§11

J5

where we have writt.en k for ty - 0: lind m for (! - ty). The solutions of this equation nrc known as \Vhittakcr's confluent hypcrg,eomcrrlc functions. If 2m is neither 1 nor Ull integer the solutions of the confluent hypcrgcomctric equat.ion corresponding to equation (11.11) :wcgivcn bycqlllition (11.G) withy=1+2m and IX = ~ - l~ 1/1. Thus the solutions of equation (11.11) urc the Whittuker functions

+

MJ.:.m(x) = xl+ me-i% IPdt -

k+ Ill; 1 + 211I; :1:),

Mk._m(a:) = x!-m e-l'" I J1\(! - k -

111;

1 - 211I; .~).

(11.12a)

(1l.12b)

Severnl of the properties of 2P\ functions have analogues for the IJi\ functions. Corresponding to equation (7.1) there is the integral formula I F t(Gt:iYi X ) = B(

1

c£,y

0:)

J0' (l-t)J'-"-lt"-le:ddt,

(11.13)

from which Kummer's relation 1F1(C'I.:;Y;.x) = e"lF'l{y-tt; y; -x)

(ll.H) may be obtuined by n simple change of variable. The unalogue of equation (6.5) is

,z {\FdIXjy;x)}=-\F'l(cr:+lj " -, y+lj x), Y

(X

(11,15)

while corresponding to the contiguity relutions of § 10 wc have relations of the type tt 11'\(IX+ 1 j Y+ 1 j m)+{Y-IXhJ.\(IXj Y+ 1 j X) (11.16) - Y 1Ft(cr:j Yj .x)=0, (x+a.:)\Fda.:+ 1 j y+ 1; ,1:)+ (y-a.:h F'da.:j r+ 1 j ,1:) (11.17) - Y \F1 (cr;+1; Yj x)=O, cr: \F1(IX+1; Yj ,1:)+ (y-2o:-.xhF\{ttj Yj a:) + {ex-y)\Ft (a.:-l j y; m)=O, (ex-y)x1Ft(a.:j Y+ 1 j .x)+y(x+y-l hf<\(exj Yj.x) y(y-l hFl{tt; y-l; x)=O.

+

(I1.1S)

(11.19)

36

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 12

12. Generalised Hfpergeomelric Series. There are two ways by whieh we llIay lIpprouch the problem of generalising the idea of u hypergoomctrie fUllction. We may think of such n function ns bcing the solution of a linear differential equation which is nn immcdiate generalisation of the e
,I."

()=m-

find notice that (8.1) is eCJuivalent to

(O(O + y - I) - .(0 + _)(0

+ Pl}y =

O.

(12.1 l

an obvious gellernlisnlion is {O({)+el-I) . .. (O+!?-I) - z(O+ot,) . .. (O+Cl..-I)} y= 0, (12.2)

where ClI' ~•...• ot..-1• 1.>1..... (!" nre constants. Furthermore it is rendily shown that this eljlllltion is satisfied by the series ~ (ot')n(Cl,,ln'" (Cl,,+-1)n. x .. .._0 (Ql )n(!.'2)n .. ,(Q,,)n /I!'

(12.3)

which is. itself. n genernlisfllioll of tile series «(j,]). Such ~enerallsed hypergeomelrlc series nnd is denoted by the symbol Hoi P.(Clt...., Cl...... ; QI' .. '. Q.), It is left as an cxereise to the render 10 show that, if no two of the numbers 1. QI' (!~ •. , '. (!. dirrer by nn integer (or zero) that the other lJ linearly independent solutions of equation (12,~) nrc a series is culled a

xl-f, rilF.(1

+ Cl

i -

Q, .. '. 1

+ IXril -

l+t?I-Q~."', I+(!.-(!/;:t).

!?/; 2- (l/.

(i=1,2, .... II).

As it stands (12.3) is a gCllcmlisllljoll or thc scries (6,1)

§12

HYPERGEOMETRIC FUNCTIONS

37

but it is not sufficiently wide to cover a simple series of the type (11.03). To cover such cases we generalise, not the differential equation, but the series defining the function. The gelleralisation of (6.1) \\'hich indmlcs (12.:J) is the series

I: n..(l

(<".(l)n(IX~)n··· (lXpl" . .'t,1l (!?t)Il(Q2),,· .. (e~)" 'II!'

(12A)

which we denote by the symbol J}F~(1X1' •.•

"'p'

[ll' .•. [lq;

x).

or, if we wish particulnrly to throw into relief the difference between the llumerator and the denominator paramclers, by the symbol

The suffix 1) in front of the P dellotes thnt there nrc 1) numerator parameters r.tl" . " Gl: p ' similarly the suffix q indicntcs the number of dcnomillator parameters. Generalised hypcrgcomctric series do nolllstlally arisc in mathcmatical physics beeflusc wc ha\'c to soh'c equatiolls of the typc (12.2). Thcir usc is more indirect. Such scrics occur normally only in thc c\'alulilioll of inl.<'grals inyolving special functions. 111 certain cases thesc serics rcducc to scrics of the type

[dl' ...,d 1J

p P

p;

Q

!?l"'" (10;

which have unit nrgulllcnt. For t.his reason it is desirnble to have information about sums of t.his type. An account of the theory of such SIlins is gi\'cll in W. N. Uniley, GCllcro!isCll l/YJlCrgcomclric Sl~rics. (Cambridge Univeriiity Press. I flll5). Ilcre wc shnll consider only one sllch calculation because it illustrates the lISC of the theorems of Gauss nnd Kummer pro\'cd above (CqlllltiollS (7.2) and (7.3) respectively). Other results of this kind lire given In examples 18 and 20 belo\\'.

38

THE SPECIA.L FUNCTiONS OF PHYSICS AND CHEMISTRY § 12

By expBnding the 31<': series invoh'oo we sec thnt s~

r(a)r(p) 1'(,) F.[ a, p, yO - F(1+1X P)J'(l+« )') 3 l+IX-P, 1+«-1';

'J

M

~

E ..-0

r(a + rllF(1 +«

"lr(P +

")1'(, +"l fJ + t1)rp +« I' + II)

_ E l'(a+,,)f'(J1+,,)I'(y+,,) -

,,-0

'11r(1+«+~1I)r(1+«

(r(1+a+,,,)r(l+a-p-y)" p+rl)r(l+ot 1'+/1.

fJ

1') F(l+ot

Now by Gnuss' theorem' (7.2) the expression inside Ule curly brneket is equal to :FdP+'I, Y+71j 1+ot+2Ilj 1) which may he written

1:

F(fJ+'fl+m)r(Y+II+III)r(I+ot+21l). r(fJ+'I)F(y+")F(1+ot+21l+m)IIl!' whence we find that -0

S

=:E i:

F(ot+,,)r(p+n+m)r(y+tl+m)

....o ...-or(l+ot+~Il+I1I)r(1+ot

P

Interchanging the order of summation 1) = 11tIL we scc that

+

S~

E

r(p+ p)l'(,+ p) ~ II-or(l+ot P ,),,-olll(p

and

(_ 1)' (- p),

1

pI

'11)1

•

so that tJle inner sum is equol to

[a, -

p

r(<<)

1)!r(1+«+1,)2

1)j - IJ ' 1+«+1'

I

whieh by Kummer's theorem (7.3) is equal to

r(a)I'(1

plf'(1 +

+ !a)

a) 1'(1

putting

r(a+ ,,) 1I)IF(I+ot+"+p)"

Now by example 9(iii) of Chnpter I (p

.

y)n!71l!

+!« + p)"

"

HYPERGEOMETRIC FUNCTIONS

Therefore ~ _ )" F(x)r(fi + p)r(y + 1»r(l + j.) . - ,:'1>1(1 +. fi y)r(! + .)F(' + j. + p)

_

T(·)F(fi)f'(y)

P

- r(l+o::)rP+0::

y)~

l' [

fi.

y;

1 l+!«.

'J '

This ~Ji'l series with unit argument eun be summed by Gauss' formula (7.2) nlld the expression for S fOllnel. It the follows that 3

'J

F' [.. fi. y; ~ 1+o::-fJ, 1+O::-Yi _ r( IH.)r( l+jx-fi-y)r(! +.-fi)r(l+.-y) - r(l+.)r(l+. fi y)r(l+j. fi)r(l+j. y)

a result which is known as Dixon's theorem.

EXAMPLES

1.

Sholl' that (I) (ii)

.""(a, {J; (J; ::) ,{'''(la,

~a

+ !;

(I - ::)-"; j; ::) - H(I

I

:)-0:

(iii)

tF,(j<:l+!, la+l; .;.; ::t) ... -

(iv)

.1,',(1,1; 2;::) _ - ;-10,1:"(1 -::);

('0)

2a:;

I

.p,(!, I'' •. "

~

I 1 +: _ -Iog--'

2::

1-:'

"

1; 1;') _ .:. K(/:);

(viii) (ix)

~')

:f',{--::

I, 1;

"

.

I; 1:') _.:. 10:(1:).

"

+ {I + :)-,,};

{(I - ::)-'" -

(1 + :)-0:);

40

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

2, By tmnsronning the cflllnUon y"+II'y _ 0 to h)'JlerJ:COlllelric rorm hy the suhstitution .: _ ,ill':, llrtl\'C thllt, ir - l:r :i: :it .:r, (i) C'O!I(II:) _ ,1-',lIu, - ill: I: sin':) (ii) 511\(11:) _ II sill: ,1-',0 - I". I + I"; :: sin':)

"nI'

th"t, ir 0 :it: :it:t, (iii) eos(Il:) _ cos (111.,-) ,"·,{jn.

-In: I: eo:s':)

+ ",in(jll,'-)OO5(:),1-·,(j-lli. 1+1";:; (i\')

lin (m) _ 'in (111.:'-),1-\(111, _Ill;

eo:s':);

I; cos':)

- llros; (lrl,,-jco, (:),1-',(i-I". 1+111: :: cos':); 3.

I'rtl\'C the relations: (i)

(~-I1)(I-~),"',(II,fJ; i';.l')

- (Y-Pla1-',(II, fJ -I: y; z) - (i'-:zl. "'(::0.- I. (U)

(iii)

(h')

(v)

(i'-P-I),"',(:z.P:i':z) _(Y_II_P_I ),F,(II,p+ I: y:.1') +11(1-:1:),"',(1.>.+ I. P+ I: y; z) _(~_P_I HI-:r),/o',(~,fl+I: y: z)+{y-II),f',(.:r.-I.,8+ I: y:.z:

(y-.:r.-jl'),/'·,(OI.jl':i';.I:) - (i' -01),1'''(:1-1. p; y: .1')-,'1(1 -.1'), />',{:J.,

P+ I: y; z):

01,1-"(11+ I: jl': y;:r) - (i'- Ih"',(OI. (I: y-l: z) - (01+ t -i'),P,(.:r., (I; i': ,f);

(I -Z),f',{II.

p: i': z) -,"',(ot-I. (I-I: i'; 2') -

(\'1)

p: y: z);

(I-P"'"

,

,

a.+I1-y-1

z,F,(ot. (I: y+ I: z):

---,F,(II,p:y~l;z)

- ,F,{ot-I. P-I: y: 2') -,f',(:z, /1-1: y; z);

fJ: y: z) (II-y);r • - ,P,(II,p-l:y:;r) + ._- ,",(11.

(I-Z),1-',(II.

,

(i)

,P,(II., 11+ I: y+ I::) -

p: i'+I: ...);

,,,',(11.. p: y: :) II(Y-P)

•

- i'-(--I y+ 1 :,",(a. + 1, P+ I:

y+~; :),

"

HYPERGEOMETft.IC FUNCTIONS

,"'1(11. P: y;:)

(ii)

• 11-/1-1- I - ,".(II-1-I,p-I;i';:) -I- ---:,F.(lI+l;fJ;y+l;:). y Deduce n simple el:jlrCSliion for lhe hypc:rgoomelric Jeries ,~\(lI. (J; fJ - I: :). 5.

H n ill n JlO!lilh'c integer.

,P,( -II, lI-I-'I; y; z) _

pro~oe

Oml

~I-)O'(I_~)r-<& r(y)

l1y-l-n)

dO

-

dz'

(zh'-'(I -z)"-)O"'),

nnd deduce Ihnt

I-I')

,PI ( - '1, lI-I-lI; IlI- I; ~

(1"_1)1--1"1"11:%-1-1) ~ , 1 .... "-1 20}"(llI+I-I-II) dJl~ (II - ) • 6.

Ir II ill II IIO.'litivc integer. nnd

("-1- 1

F 'I 7.

2'

l~lIhilJlilih

(i) (ii) (iii)

_...!..) _ (_l)o:r~+l

11+2. I' ~ ";c'

II!

d" (

dx'

I \/(;c'-I-I)

1J6(.rO',F.(II.

.

fl: y: kZ)}_t;U'l-l. 1o',(lI+ I; flo y; k.r) where 11';60;

ll(l. y-l),Io\(II, (J; y; z)_ J:,.l-I(I-W-.l-I,f'I(II, p; I; zl)dt,

Iz I <

I, ;, > 0, }' - J. > O.

P>

0,

II-I:).... JI" .

/I: 2/1;:) - ~~-IB{jJ. (J)

0 (511I9')',1-·

[ ( I -I- COOS/P}....] {I _ C0059'}- 119',

+

where ,; - :/(:: - =). Dcduce that

9.

)

the following formulae:

Prove IIll1t if

,1-\(11,

I. prove IImt

,1o\(lI; fI: lI+P+{I; z)x,Io',(y; J; y-l-J-{I;;c) - ,1·\(lI-I-!.', (J-I-{I; lI+fJ+{I;:I;) X ,"\(Y-e, J-{I: ,+6-{I);

whcre

8.

Iz I >

I'nwe thnt

JI"

;Tr(,.-I-I) o IlOsIIlOcos'OdO_ :lu'J'(jll+lm+l)r(in

Im+l)

42

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

lind c\'nluntc

l,

J

o CO! 2mO sin ft 0 dO

where III ill n llOSith'c intcger. 10. Sehrodingcr·... c'luntion for Ihe rotation of a symmetrienl·tofl molecule i.'l

I

,(.

aM

sin

'OJ

lo"lf'

+ sin'O

sma M

Ila'

A)

+ ( col' 0 + C

2eosO O"lf' IlX" - siu" 0 IlXllrp

Il'""

where A, C. 11', I, nrc ooustanl!l. Show thnt il the form If' - cl_-j-t(l - o7)llft-..I;rllft-..llJo',(ex, (J; whrre fl G:: m. or - HI root.s of the t1lulltion

-

cosO), i'''' 'I If

: ' - (21/

II.

8:r"AIV

+ -,-,,-If' _0, posse:ucs solutions or

In

+

r;

07),

1, and ex. {J nrc the

8:r"AII'

+ 1): + CUI + n --h-'- -

O.

Prove Ilmt: (i)

''''I(ex; 0:; ;r) _ c",

(ii)

,I",(er.

+

I; 2; ;r) _ (1

(iii)

,1'\(1; .:.; -

(iv)

,1'\(a.+1;;,;;r) -

(v)

(vi)

12.

,1'\(_1; I;

.1'1) _

+ :) to.

V· crf(r). _.•

-;;-

Y

-zl) _ c-" -

x ft ,I'''(Il; n+1; -x) -

v'no7err(:c).

"f:1ft-lr'tll.

l'ro\"e lImt tile erlUlltioJL ;)'1'

1 0 I'

or" - k "Ft'" pos.~ ~olutiol\s

of the type

,,_ C/",P, (- III; when'

III

,

,l'\(er.;;,; x)_ -,F,(a.+I; ;,+1; 07).

nud C are const.... nl.1.

HYPERGEOHETRIC FUNCTIONS 13.

43

51101" thnl the Schriklil lgcr eqUAtion

P'lI+( k'- ~)lI-O 1IOlutiou of the rorm

POSlK'SSC S II

til<

(_.i!.: '!k

1"\

ik=).

1: ih _

14. The SchrOdi ngcr equnllon gO\'ernin g the rmllul wnvc (unction s for positive energy Itutcs In II COUIOlIlU licit! b

(dL) + [a.,"" . ='t') 1)] 1._0. - - (11-- - 11(11+ ---

I d rrd,

d,

I.'

r

"

Show tlmt it possesse s a solution L_r~~' b,f',( i:z+II + 1:211+ 2:

where k' _ a.,'mll-/h. a _ 15.

-:likr)

4."m=·~'Jk.

Show thnt the equation

d'y .ly+ ( "" - + -1 d.l:' 'Z liz

poSSCSSCI

'!.1Il{J

II'

z

Jy_O

01"

n solution

y _ zl-t-I'l f',llu + • - i{J: 2im.f) 8l1tl hence thnt n solution or C(IIHltioll (LOc) i"

n _ ~i~c~hJo'UII + I:

16.

/I

+

I: 2i,,,~).

Show lhut (il

(ii>

IIo

Zl-I(I

a.: ltZ] ,,-,

-z)..-I.F. fa lO •. " lflh.··.{ J••

_ /'(1 ~ • III I ..1F •• 1

r

Z'-I(l_z ),"-I .f'.

[ai, ....a,:

-;r')--I.""

[III'" .• 11.:

Pl' ...• (J. I/(! -

J

!-' •• ,

.. I

[lIl' ...• PI"'"

a.,

I

+"',

III:

]

i]

P•• I + III

I~J
) .111

"I'~'

~:, ,iz

) ,1/1

(I

1

PI'" .,P.' _ n(1

r

II

I'"

o

(iii)

[P"" .. a •• /, "

•• ,

•• \

f:r.I'·· .,IIt.> Ill; I] LPI.... 'P•. 2m;

44

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

17.

PrOWl tlmt, 0)

fa> .1". rCliI."" IX.: o LPI' ' ,.. P.,

birJ

(,-":1',,-1


FIJI)

LPp" "P.:

- -;;;;- ."P,

(ii)

(Iii )

foa>

.1".

ra"

,tt•• ±/;'.l'J e-.'·';t:I'-' liz I'U/I) [P" ',"Cli., --P

LP"

, ft.:

Q

2]11' " "

.. ' . "".:

By e
IS,

(1 - x)a.-IJ-~ ,f\(a, {J: y; xl - ,P,(y -

prove

S:Hl.lschut·£'~

theorem

a,

p,

-II: 1

•f', [ Y, 1+o:+{J-Y-1I

J

Cli,

Y - (J; y; z)

(y-tt) .. (y-P) •

(y). (y

P)..

Cl

lienee jlrovC that .",

,I<\[a, {J: %

] _ (1_.:z:t
-

(l-%)'J

irl.:z:I<:J-2,f~

Show lllflt

19.

,I<',(tt, (J: 1 +tt-P; :)-(1-:)-,f\(ja., I+la.-I/: I +a.-P. C) where C 1 + tt - P; 20.

- 4:(1 - :)-'. lienee deduee the vnlue of

-

I) (rom

,J.'I(a.,

Pi

theorem.

Sho\\' thut

.[0. II,P. II.Y' 'J -

If.

Gllll~S'

.t

J'(J)l'(ll)J'(a)

1'«1) r(a+{J)J'(a+y) IJo'.

[J-:l, e-IX, a; 1J a

+ p, a + i'

+ll-IX

where u_J -{l-y. lienee, usillg Dixon's tholorem, prove

\\'lltsQn'~

lI'CorCIIl:.

HYPERGEOMETRIC FUNCTIONS

"

Ct. fl. y: I ] I'mr
WIlI.sOIl·~

theorern. deduce Whipple', theorem thllt, ir

IX+P-I, 1l1llI,+"_2y+ I. •

,. [<1. p, y: '] :I/'("ll'(1I1 1 6. t: - 211'-lni2+!6In~+idr(ifJ+i6lrUfJ+ltl

CIIAPTER III

LEGENDRE FUNCTIONS 13. Lcgcndre Polyllomhlls. If A is n fixed point with coordinntes (0:, fl, y) nnd P is the mriable point (:r, y, :), thcn if we dcnotc the distancc A P by R, we have

11'

~ (x -

.)'

+ (y - P)' + (, -

y)'

Furthermore, we know from elcmentary eonsiderations that 1 ~~n

is thc grnvil.nliollal potcntial at the point P due to n unit mass situatcd at the point A. and that this must be n particular solution of Laplacc's equation. In some eireumstnnecs it is desirable to expand 'I' in y' :2)1 is the dislnnee powers of r or r l where r = (r

+ +

p

,

11

01 0

" Fig. 4 A

of I~ from O. the onglll of coonlinutes. This expansion can be obluint:d by the usc of Tnylor's theorem for functions of thrce \·nrillbles but it is much marc suitnble to

'"

113

"

LEGENDRE FUNCTIONS

introduce the angle 0 between the directions OA. OP (ef. Fig. "') and write J'l:! = ,.J + a~ - :lar cos O. The c:-.:prcssion for tp then becomes 1

1J' = V(ll~ _ 2M!1

+ r2)

(13.1)

where I' deflotes cOS O. and this enn be expanded in powers of rIll whell r < n and in powers of nlr when r> fl. If we denote by J\(I') the cocHicicnl of II" in the expansion of (1 - 2,uh It~)-. in ascending powers of h, i.e. if

+

1

V(1

•

•-I' , + I') = ,,-0 1: 1'..(p)II".

(13.2)

I

j

then the potential function (!fI.l) con be cxpl1l1dcd in the forms

~ ~ a

.-0

(!-)" P ,,(/1), a

.!- ~ (-,,-)"/·n(J1).

r

<

fli

(13.30)

r>

a. (13.31» r ..-o r II is clenr from the definitioll (2) that the coefficients P,,(/I) firc polynorninls ill It. The first one or two cnn readily be calculated directly frolll the dcfiniton. By the binomial theorem we have (1 - :!ph

+ ,,~)-l

= 1+(-IH-2J1h+h!) = 1 +,1I1J+!C3/12_1)'. so LhnL

+(- !~(- ~)(_2.1I1J+h2)2+ ...

+ !(5p 3_3J1)" + ...

p 0(11)= 1, PI (ft )=/1, P 2{,1)= ~ (:3/12_ I ). P3(/1 )=~ (5,113_3fl). (13..1a) We shall show below that. in the J.:cnernl case P ,,
48

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 13

The expression for the gcneml polynomial P.. (p) can be dcrh·ed by the method employed 10 obtain the simple expressions (13.4a). Expanding (1 - 2Jlh 1,':)-1 by the binomial theorem we hnve

+

(1 _ 2/Jh

~

+ h2 )-1 =

r-I)

(!), (2"ft _ 11 2)'

r!

and the coefficient of It" in this expansion is the coefficient of h" in the e..."pansion

i: ....0

i

(!), (2/1h-I,')' = (1-)..-. (2/111-h2)f1-' r! .-0 (II-e)! ~ (!).. (2/Ih-It')"-, =~(-I)'(l·:"I), (11 (l)! '

since by example D(ii) of Chapter I

1- I)'W.-, ~ 1,1'),:)" Now the coefficient of l,n in the expunsion of ( - 1}4l (2/111 _ h!)"-'

(11

!?)! (2/1 }..-!q e!(u 2(l)I'

and, by the duplication formula for the gamma (\lndion, II!

(11

2(1)1 2

so thnt

__ =l1_

rUI/+U

r(ill+!

F(t"+1)

(llF(!,,,+l

Q)-(!-tll),(-trll"

E

(2-)',

p (p)= (l).. (2,u)" (!-!Il),(-!Il), .. II! ,..0 e1(! Il), ,u'

a result which may be writtell in the form P .. (,u)=(2/1~t~)"tPl(!-!Il,

-;11; i-II; I~'}'

(l3.-1h)

113

"

LEGENDRE FUNCTIONS (13.~)

Putting It = 1 in equation ficients of h" we find that

nnd equating' coef-

P ,,(1) = 1

(13.5:1)

for nil vnlnes of'II. Similarly if we put I' = we derive the result

p.(- I)

~

1 in (13.2)

(- I)"

(13.5b)

which is n particular case of the l"CSult 1'..(-/1)=(- 1)" P ,,(It).

(13.6)

Equution (13.4) gives P ,,(cosO) as n polynomial ill cos 0 of degree '/J so that it should be possible to express l',,(cosO) in tenns of cosines of llIultiples of O. Instend of attempting to do this by substituting the appropriate expression for cosrO in (13 .... ) we begin afresh with the definition (ltl.2). Writing (1 - 2 cos Oft Jt2) in the form (1 - e iO h)(l _ e-iOh) we find t1utt

+

•

1; l\(cos Olh" = (1 .-
lIe I6 )-I(1 _ hc-iO)-i

_ ~ ~ F(r+! )r(s+t) h'+> ei1r-o)O

-r~.-o F(!)rU)r!s!

Equating the coefficients of h" we find that P ( 0) _ ~ F(i+r)r(!+II-r) fl:r- .. " .. cos r(-!)r(i)r1(11 r)! (; .

-::.0

Using the duplication formula. we see that F(-!+r)r(i+1I-r)

rmrm

1 (211-2r)1 (2r)!

2""

r!(11

r)!

so that P (

n COS

0)- 1 ..,'!, (2/1-2r)!(2r)! jC~r-"llt - 22" r:'o (rl):{(ll r) IJ2 t: ,

from which it follows immediately that

.

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 13

50

HZ,,)!},

1;::1 (ZII-Zr)I(2r)!

P~.(oo50) - 21"(111)' + 2100- 1~ (rIPH'1 r)!}2 cos(2"-2r)0. (13.7) nnd __ 1_ ;

P~"+l(cos 0)- 2'10-1

(211-2r)1 (2r)1

'-J_O

r::'o (rl)l{(lI r)W 005(_11 _r+ 1)0. (13.8)

1·0,--------------:;',1

o

0·5

0·25

1l=.~

Fig. 5

Vllrilltion or 1'.(,1) wilh II.

From thcsc lust two ccpmlions we mllY dcri,·c a general rcsult of some imporlancc. We may writc

P ,,(cos 0)

•

E cr ooS(1I - 2r)0.

=

(13.9)

~O

whcre p = ill or In particular

~11 -

so that from (ta.5)

~

according ns 11 IS c,·cn or odd.

113

LEGENDRE fUNCTIONS

1=

• I:c,.

"

.-0

Now. from equation (13.9) we have

I P ,,(cos 0) J ::s;; Z Cr .-0

nnd therefore (13.10)

0·5 p ..(cos 0)

Fig. 0

Vodation or 1'.(,"OlI II) with O.

The \'ndation of l\(p) with Il for fl few values of " is shown in Fi~. 5. Since, in O1Ol>t physiclll problems, the LcgclUlrc polrnominl invoh'cd is IIsually P ,,(cos 0) we have shown in Fig. a the \'nrintioll of this fllllcLion with O. Numerical values may be obl:lincd from Tubles 01 As.tocit/lnl L,gmdre f',mctiOll8 (Columbia University Press. 1!H5).

S2

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 1-4

14. Recurrence Relations for the Le~cndre POlynomials. If we differentiate both sides of equation (13.2) with respect to Ii we have (1

,,- Ii _ ~ 11 In-Ill ( 1 ~Jlh + 112)"11 - "":.0 ~ "It,

which mny be written in the form

•

•

• ..0

..-0

()I-Ii) E II" P .Vl) = (1-2Jlll+1I 2) :E 11/i1l-! P .. (.11).

(1-1.1)

Equating oocfficicnls of h" we have

,iP"Vi) - P _1(/1) = (tl+ 1)J~NH(P) - 2n,IIP "
which reduces to (-11+ 1)p,,+1(/1)- (211+ 1 )/IP,,(JI)+11 P ..-1(") = O.

(14.2)

This relation has been proved to hold for IJlI < I but since the left hand side is n polynomial in I', it must. hold for all vnlucs of II. On the other hand, if we differentiate both sides of equation (13.2) with respect to Jl we obtain the relation (1

11 -~'''P' ( ) "/ "II, .1" +/'1"' I I - ...... -0 I

Combining cClllnl iOlls (I·L I) and (I"i.a), we have

so thnt e
P' .. _l{U) = I1P..(U),

(HA)

'"

LEGENDRE FUNCTIONS

53

and since each side j<; a polynomial in I' this relatioll holds for nil values of I'. If now we differentiate equation (1 t.~) with respect to II. we obtnin the relatiOIl (1I+I)P'n+1C!1)- (~1l+1)P..(I') - (:lIl+l)pl-·",,(Jt)+t1P',,_,(/t)=O.

(I;J.5)

Eliminating' P',,(u) from (1.IA) and (l.k5) we see thnt

("'.0) Subtrncting (U.,q from (l'l.6) we outllin the recurrence

relation

The differentintions with respect La 11 IIl1d II IInder the summntion sign is justified by the fact thnt the series on the right-hand side of Ctluntion (13.2) is uniformly collvergent for nil rc:l.1 or complex values of h and I' which satisfy the relation Ipl ~ 1, III I < "\/2 - I. 15. The Formulae of Murphy and Rodri~ucs. From the expansion (13.2) it follows immediately lhut

d'

~

dr

1: II" -/-r P .. VI) = -/-r (1 - 2p"

.. -0

f

It

(

=

Substituting thc '·ulilc

i: p~I(I)h" =

.-<>

It

+ h )-1

w"r rerr(!+I !) I~

2

(I _ "ph

-

+ "2)-r-l

= 1 we sec lhat

+ t) hr(l _ rm

2r r(r

+

")-('-11

+ '.!r + .r)!l'

= .•rr{r t) p V ~(I r(t) I,~ 1'(1

where 1.J~I(/I) denotes drP..(/d/d/~r

+ 2r)81

S4

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 15

Equating codfieient5 of II" we see that p~rl(1) = 0 if 11, as is obvious from the fnetlhat I>,.(p) is a polynomial of degree 11 ill I', and that

r>

pl')(1)- '1,F(r+~)

"

- -

r(j)

1'(1 + n+ r)

r(l

+ 2,)("

,)1

From the duplication formula for lhe gamma function

2-r(, rU)r(l

+ !) = I 1 + :!r) rl2' =(1-),2"

and from example 9(iii) of Chapter I, r(1+11+r) 1'(011

r)l

=(-lnll+ 1 ),(-n),.

so that /-.frI(I) = (_ly(Il+ 1M-II),. .. (1),2r

(15.1)

Now, b)' Taylor's theorem P ,,(It) =

t

(J-l -I J Y p~I(1) .

.-0

'

Substituting the exprcssion (15.1) in this expansion wc obtain thc relation

P .. (p.)=~(-1I),(ll-:--1),(1 ~

(l).r.

... ~' _ -J

whieh gh'cs Murphy's formula (15.2)

for lhe Lcgcndre polynomilll P,,{.II). If 1I0W we put ox = I ill c."nrnple 5 of Clmpter I r wc scc that equation (15.2) is equivalcnt to I

tI"

P (p.)= I','_l)", .. 2"111 lip" v

(15.S)

LEGENDRE FUNCTIONS

§15

55

which is Rod&i~ues' formula for the Legendre polynomial. Rodrigues' formula is of great lise in the evaluation of definite integrals involving Legendre Consider, for instance, the integral

polynomials.

(15.4) Dy Rodrigues' formuln we lllay write this integral ns I ""I _ II

f1

dn j(x) /,,(£1_ 1 )"lIx, -I (X

and nn integration by parL<; gives

I

n [d - 1

J'

2

I

2"'1 [ dX"-1 (x _1)" _I - 2"111

fl_/(x) dX"-1 (/"-1 {(X2_1 )"}dx.

The square bracket vanishes nt both limits so that we have

fl

1

I~ - - , 2"11.

_I

d n- 1 /,(x)-/-, f

x"-

((x'-l)"l't.x.

Continuing this process we find that (I5.5) For example if I(x) = P ",(x), so I = O. In other words

III

<

II,

IIIII(X) = 0 nnd

,

J_tm(x)P,,(x)r!;t:= 0, If j(x) = P,,(x) then

(2/1)1

=

2"11['

(Ill =F'11).

(15.6)

SS THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 1S

nenee

:'lInking lise of the duplicIItion forrnulu for thc gllmmnfunction wc ClIll rcducc this to the form

I'(Po(x)j'"x=., -I

"+-

811

I

(15.1)

•

A con\'Cllient WllY of combining the results (15.G) and (15.7) is to write (15.8) where 6... is the Kronecker delta whieh tnkcs the value o if 111 i=·11 lind the YIIlue 1 if m = II. Similnrly if I(z) = ,2:'" wherc III is 1\ positive integcr, thcn

r(m Illll(x)

=

[

nnd hencc, if III

I

I

It'''P.. (x)dx =

_I

1'(11I

>

+ 1) .m_o + 1)

O'l

if

/II. ~ I/.

if

'III,

<

'II,

II,

r(m ,!1Ir(m

+ 1)

"+ 1)111

II a:-II(I_z1)lldx. -I

H 111 - /I is an odd integer the integral on the right is 7.cro while if 1/1 - 11 is lin c\"cn integer it hIlS the value

so lhnt, if

111

is lin integer.

116

if

0, l

r

x"P (z)dz-

_I"

If

111

"

LEGENDR.E fUNCTIONS

=

'II

111

IrUIII-!tl+l)

{ 2"

(

111

)'r(1

".

I

III

< 11,

.

"

,),lrm-ll:::::::01SC\CIl, (15.9)

~m+ _"+~

0,

ifm-11>Oi:;odd.

Ute result is

f'_tX"P,,(x)dx= 2" f'_I(I-orZ)"dz 1

1 r(lll'(n + 11 = 2" -I- S,)

hI!

which, on account of the duplication formula, is equivalent to ( 15.10)

16. Series of Lc~endre Polynomials. In certain problems of polcnlinl theory it is desirable to be able to express n given function in the form of 1\ series of Legendre polynomials. We Cfin readily show 1hn1 this is possible in the case in which the given function is n simple poly< nomial. For CXlllllplc, from the Cfjlllllions (13.4a) we have 1 = 1'",(/,1,), l.t = P I (,,),

+ 'frP2(/t) = 1!PO(/I) -I- 5P2{j1) !t3=~/t + ~ 1\(11) = {tP\(/I) -I- %1-'3(/1)

/,2= ~

so thnt flll)' cubic as thc scries

Co./I'J

-I-

CIJl~

-I- c:." -I- c:J cun be

2~O J>3(P) + 2~1 I\(JI) +.( ~o + ~) P1(.u)+

wriltCIl

C;I + es)

1'0(,1)

It is ob\'ious that wc could proceed in this WlIr for [I polynominl of lilly gh'CIl degree" though if II were large the nrithmctic irwolvcd might becomc cumbersome. Since

56

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEl'lJSTRY § 16

Pn{jl) is a polynomial of degrec II in It, it cmcrges as n result of an cxtcnsion of Lhc abovc argumcllt, that allY polynomial of degrec II ill It can be cxpressed ns a scries of the type

,

,...,

LCrPr(/t)

(-l~/t~l)

(lU.l)

The problem which no\\' nrises is that of exprcssing' (lily function !(p), defined in the inten'al - 1 :s; ,t:S; I as a series of Legelldre funcliolLs of the fonn (16.2)

If it is assumed Lhllt the infinite series (10.2) eonvergcs uniformly in the rnnge (- I, 1) to the sum !(/t), we may multiply the l.cnns of the series by Pn(/t) and integrate term by term with respect to O\'CZ' the rnllge (- 1, 1) to obtain the rt::llLtiOll

.

,

,

fj(/d P n(ltlc1/1 = ;/r J_tr(/t)PnV.t)dlt nnd by equation (15.8) thc sum on thc right hnnd side is equal to

..

~

()

, --- b

r_O r:!I1+1 r,n

<>,

- -,,-";-'-"-c 211+ 1

which shows that the series (I U.B)

converges uniformly to the sum !(p) in the range (- 1, 1). Thc series (16.3) is culled the Le~cndrc series of thc function !VI). Wc shall not discuss here the conditions which must be satisfied by the function !(p) if this series to be uniformly COll\'crgenti for such a discussion the rcnder is referred Lo ChapLer vn of E. W. Hobson, 1'he '}'hcory o! Spherical mlfl 1!.'llipsoitlal llarmonics (uunbridge University Press, lfllJl).

116

LEGENDRE FUNCTIONS

"

The possibility of expanding n function in the form of n series of lype (11I.2) is n consequence of the relation (15.6). Tn the theory of special functions we frequently encounter sequences of functions 9'1(.%)' 9',(z), .•. , 9".(z), ... which have the property

f••1p",(z}!P,,(z)d.r =

0,

(III

#- Ii)

(IG. q

We then sny thnt the funclions 9',(%), (r = 1, 2, ... ), form nn ortho~onal sequence for the inLcrml (a, b). H, in nddilion, tJle functions urc such thnt

f••{rp,,(x))'tlx =

1,

{I 0.5)

for nil vnlucs of II, we sny thnllhc fUllctions of the sequence nrc normalised, nlHl form an orthonormal set. Given II set of orlhogonnl functions it is obviously 1\ simple mutter to construct n normalised sequence. J'or example we sec from equlltion (15.8) that the sequence of functious 1\(x), (II = 0, 1, 2, ... ) is orthogonnl bllt not normalised. By lilultiplying ench function by v(u +!) wc find that the functions (Ii + !)iP.. (x) (10.0) form 1\ sequcncc of normaliscd orthogon:l1 fUllctions in the illtcrvnl (- I, I). There is another property of import:mce which such n sequcnce of functions may possess. If there is no integrable function P(x), different (rom zero, sueh that

f••v:(x)/P,,(z)tlz =

O.

(16.7)

for all "nlues of Ii wc say that thc sequence i<> n complctc orthogonal sequcncc. It may be shown in thc casc of thc functions (16.0) I) by coll5idcring thc Fouricr coefficicnts of V/(x/:T.)P..(:r;f1t) in (- a, :it) that if (lO.n I) E. W. Hobsoo, 0". cit. p. MI.

60

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 17

holds this function is n null-fullction nnd hence that yr(:a:) is n null-function ill (- 1, 1). In otherwon:ls it c::m be shown that the functions (lO.6) fOfm II complete sequence of normalised orthogonal functions.

17.

Lc~cndrc's

Differential Equation.

]f

we write

II = !II' -~1)" then it is readily shown thnl (1 -l(2) (/fv , I'

+ 2, t1

/1)

= 0

+

llnd if \\'c differenliate this equation n 1 times using Lcibnitz's theorem we find that dr
d~' -

shows that d"(Jl 1 -

2p ~l

+ n(1I + I)} ~;) =

0

1 )"/d/j" is n solution of the differential

cC]lmtion ([ly

rIy

+

+

(1-/1 1)-/2- ~Il-d 11(11 l)y = 0 (17.1) (It P so thnt we conclude from nodrigucs' formula (15.3) Lhnt when 11 is nil integer Pn("l is one solution of the equution (17.1 ). This equntion, which we shull now consider in II little more dclnil, is called Le~endre's differcntial cquation. We saw in exnmplc 1 of Chaptcr I holl' sneh an equtllion nriscs in thc solution of Lnplncc's equation whcn solutions of the type R(r)8(eos 0), (i.e. III = 0) are sought. It is olwious by inspection that the point p = 0 is nn ordinar)! point of the equation (17.1). Writing the equution in the form

+

cFy '1}' dy lI(n 1) VI-1)2-.. + ( / 1 - 1 ) - - - VI-I)y=O (lp~ u+1 dll /,+1

LEGENDRE FUNCTiONS

§17

and observing' thnt in the notation of equation (3.1) wilh (1=1

1'(")~

1

+'I(,,-I))~1+11,,-1)-tll'-I)•....

1+~

I-l

~

1

It-l

q(/I)=-!Il(Il+1)1+HII ~

1)

- lu(u +

ll{(,,-I)- ,I" -I)'+ ... j

we sec that It = 1 is n regular singular point with indicial = o. Furthermore in the notation of equation (.1.2)

CqllllUOli (l2

iX(lI)

,

- 2}t =-.. l-W

R( )_1l{1l+1)

I'll

-

"

1

,

""

so that liS II -+ CO

RI) 11(11 + 1 ) ,,1''''''''.. , ,lI"

showing thnt the point It = co is n regular singular point with indicial equlItion {Q - (n I)} (/? on) = O. We thlls sec that the cqunLion is defined by the scheme

+

+

-1

y=P

0

\

o

00

I

n+l

0

-'II

0

) j1

•

117.2)

If, however, we put x=!(1-,Il)

in equntion (17.1) we find that it reduces to the form (Plj

liy

IX"

fX

X(1-,T)-/',,+(I-2x)-/ +'II(Il+1)y=O,

+

(17.3)

which is Cfl'Jalion (8.1) with IX = 11 I, fJ = - nand y = 1, so that the scheme (17.2) is equivalent to the scheme

6'2

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 17

!I=pf~ tl~l ~

10 -

II

0

i-!,u!.

(li.4)

It should be noticed that the values; along the top row are those assumed by ! - !11 1I0t by /,. We consider first the solution corresponding to the singulnr point at infinity. We write

•

Yltu) = /I" £ 1:./,,-. -0

which on substitution into (li,l) leads to the recurrence relation

+ 2)(11 - V + 1 )C....l

+

= - v(21l 1 - I')e. = I we obtain the solution 11(11-1) _" JI{ll-1)(fI-2)(II-3) "_1+ ., (I' )-,,",I 2.(2/1-1),," 2A.(21l-1)(2/1 3) ..• (II -

l'

On Inking

('0

-+

=

I

"

I" (1+ HI/Hi-in) 21.(!Il)Jt~ + (-,,, )(-1"+ 1 )(,-1" )(,-1"+ 1 ) .!. + ...

which

11m)'

1.2(! Ill(! "+1) be written in the form

ydp)=/t rl :F.(-!Il,

!-!1Ii

1-11;

!t~

I~:l)'

1

(li.5)

1\150. if we write for the second solution

we find that

(17.0)

1 17

LEGENDRE FUNCTIONS

63

provided " is nny number other than a neguti\"c integer or hnlf n Ilcgnli\'c integer. These solutions IIfC \'lliid for nil '"altlcs of 11 for which the ,PI series have n meaning. not only for integral ,"nlucs of fl. If II is IlIl integer the series YI{!') terminates - in olher words, yIC/I) is n polynomial of degree II in II. If we multiply this polynomial by (2/1) ! 2"(11 !)2

we obtain the Legendre polynomial of degree n (equution (l3Ab) above). Onlhe other hand the series for Y2(/l) docs not terminate when /I > - 1 so there is no point ill restricting n to be nn integer. This series solution when multiplied by n fUclor

rCi)!"" + I) 2"+lr{1l

+-})

gh'es the function _r(Vr(f1+1)

Q,,{Jl)-~"+lr(II+~)"

_'I-!

,

(

.

3. I)

2"\ ~1l+!,!"+1·1l+:r'/12 (17.7)

The fUllction

is n solution or the Legendre cqulIlioll (li.4) even when " is not nn integer nud it reduces to the Leg-endre polrnominl when '11 is nn integer. We shall continlle to denote it by P ,,(/1) bllt when 11 is not nn integer shnl! rerer to it as the Le~el1dre functIon of Ihe first kind of de~ree IJ. The funcLion Q.. {p) defined br cclunt-ion (IG.i) will be rererred to ns the Legendre function of the second kind of degree 1J; even whell 11 is 1\11 integer it is not n polynominl.

64

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 17

With these definitions we may write the solution of Legellllre's eqlHltion (17.1) as

(I,l I > 1). III some problems we know that the solution should ben JlolYllominl ill/I; in that case we lUust take the solution to tile form y= AP,,(/I). The vnrinLion of Qn(/l) with II lTIay be computed easily from equlltion (li.7) when I! > I. Tablcs calculated in this wny lire containcd in the \'olumc quoted at the end of § 13. The following Fig. 7, whieh WIIS prepared from these tllbles, shows the vnrintion of Q.. (/l) with /l for a few valllcs of II. 0·5

U··j.

11=0

o·a

I/=J

Q,,(fl)

11=2 11=3

n·2

(l.

I

,

0·0 1 Fig. 7

-1'-

"

.,

Variation of Q,,(/t) witll I'.

Since Legendre's equation hilS 1\ regular singular point 1 we may on the bnsis of cquation (3.8) take the

lit II =

116

6S

LEGENDRE FUNCTIONS

second solution ofLcgcndrc's cquntion to be proportionnl to

p.V.)1og V. -

1)

+ ~ <,l!' -

1)'.

The coefficients cr can now be ohtnined by substituting this expression into the differential equation (17.I) and C
I

P ,,(e)d$ _I It ~'

I

where III I > 1, nnd 11 is n positive integer. Expanding the denominator by the binominl theorem. we ha\'c, for the \'uluc of the integral, the series •

1

~ It ...,

-0

I' " P.('ld'. -I

From C
+

• 1:

r-O

It

1 "+1+2'

I' -I

~"+'2rPn(~)d~,

which, by thc snmc formula, is C
11."+1

(11

+ 2r)!F(r + !)

.:0 2"(2r)!F(1I + r +.Q) Q)

(')'

II.':

Now from thc duplication formula (11

+ 2r)!P(! + r) rein + i + r)re!1I + 1 + r) 2"(2r)!

-

so that thc series reduces to

r!

66

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 18

_i_ ~

F(!Il+I+r)r(tll+ l+r) (2..)' r{lI+

lIn+! ....0

-fr +r)r!

p2

which is equul to r(~I+!)r(!II+I),( ~ r(11+~)/In+1 2 P \ tll+~,

,,3.1)

111+1, Il+:r, Jl2

•

Noticing that

PI,. + ,WI", + 1 lIPI" + t)

+

+

~ pmp(·" 1)/2"1'(" !), and comparing the result with equation (17.1) we sec that this series is merely 2Q.. tlt). Hence we have shown that if III I > 1

Q"II')~'f'

P"(i;,,<,

-1/ 1 -

(18.i)

..-

Il result which is known as Neumann's formula. Equation (18.1) holds not only for rcal nducs of.u greater than 1, but for nJl vailles of II which nrc not real. For this renson equation (18.1) may be rcgnrdcd as defining' the second solution, Q,,(/t), of Lcgcllclrc's equation. Certain related formulnc, due to MncRobert, follow readily from this result. If Ifl I > I and 11/. is It positive integer then l

fl"'Qn(fl>-tf

r"P"~)d~=!fl fl,m_~'''Pn{~)d~

It -1 It - ~ unci the integral on the right is equivalent to the finite sum _I

"I-I

tE

....0

I

ftm-I-rf ;rp,Mld;

(18.2)

_I

Tfll~:S;;:'/I

it follows from equation (15.0) that eaeh term of this series vfinishes so that we have 118.3)

provi<]ed ?Il, /I. nrc integers find III :s;;: II. On the other hanel if ?Il = It I, the series (18.2) reduccs to the single tcrm

+

LEGENDRE fUNCTIONS

I"

•- f'-I c" l'"(cW

~

so that

JII+1QI ..

IIV-

"-'--""",,,.,., (211+1)1'

P (') ""I I)' )=*f" ." d;-+·(211+ - _1 f1'; 1)1 I

I

~

"

""(0) I)'

~"+l

Tl

.

(18.4)

Now if 111 is nn integer P ...
1/1

'f'

( )'1 "(I I~ },,~!I I:!

1'.(')/'.(i) (18.5) : (I' ... 1/ .. Similarly from cquillion (18.-1) 1I1ld the definition of /',,+d,J) we hnve the formula P .+1(plQ.. Vt)

=!

-I

(') f p . I I(,)/' Ii; + -+l' .. I

I

+1

:""

-I

/I

(18.0)

If we replace " in CflunLion (lS.5) by ,,+ 1 and til by '1 :lIld subtract from eqnation (18.6) we obtain the relation PHI{/I)Q.VI) -

P,,(JI)Q,,+lVI) =

1

'1+ l'

(18.1)

Other formulae of n similar nature nrc contained in ex. ~H below. 'Ve shnllnow make usc of !\clllllnnn's formula to derive t he form of the second solution of Legendre's cquiltion ill thc neighbourhood of the point.'l II = ± 1. l"rorn (18,1) wc hnve the rcsult. that. if I'. >1, 11+ 1 (18.8) Q,,(J-l) = !/\(.If)log,ll_ 1-11',,-1(11). where 1V"_1(;l) dellotes the inlegrnl

! fl P ,,(/l) - P ,,(E") dE. •

_1

II

~

Now when Ii is nn integer, P ,,(II) is n polYllomial of degree II in I' so that {P ,,(p) P ,,(;)}/Vl is a polYllomial of degree 11 - I in p. lienee 11',,-1(/1) is a polynomial of degree" - 1 in /1.

n

66

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 18

If we substitute from (18.8) into Legendre's equntion (17.1) we rind thnt 11'..-1 53-tidies the difrerentinl equntion

+

(1 - /I:!)IV;~I - 21111'~_1 11(11+1)1V_I = 2P~VI) (18.0) Now sinee 1I'1I-1Vl) is n polynominl of degree Jl - 1 in II we mny write (ef. § 10 nbo\'e)

.. . .-,

11'.._1(/1) = I: l:r/)r(,U)'

,

Using the fnct lhnt

,

/',,(11) = ~

_0

(211 -

'~r

- I)P n-21"-1'

where l' = t(1l - I) or til - 1 according liS Il is odd or C\'CII (which follows from equntion (I.Ui)), and the result (1 - /t:!)P;' V')

-

211 ['.(It)

+ 11(11+ 1)P.(/f)

+ +

= (Ii - r)(11 r 1 )/'.(.,,), we find, from equalion (18.0), thnlc._ 20 = O. (8=0, "',1') Bnd lhat 2/1 - "8 - I cn-:!I-t = (28 1)(11 8)

+

Substituting these vnllles in (18.8) we obtain the formula "'

11+1

'!,

:!II-'~$-l

Q,,(Jt)=·P,,(II)log/(_I-.~(28+1)(o'1

8)P"-~1(!l).

(18.10) Another expression for Q,,(J1) may be derivcd from thc fnct thnt both P,,(J') lind Q,,(JI) nrc solut.ions of Legendre's equation (17.1) so that

Q.(p) dd

HI -1,')P;(j,)} -

J1 which is Cf:luh'alent to

p .(1') _,d ((I - u')Q; (j,)j = 0 (/1

•

d [(I-I,'){Q.(j,)P;(p) - 1'.(j,)Q;(p)}] = 0, dI' showing that (l-I,'){Q.(P)l';l!')- 1'.(,<)Q;(j,1}= C, (18.11)

"

LEGENDRE FUNCTiONS

I"

where C is a constnnt. Now from equations (17.7) and (17.8), we cnn rcudily show that for large ,-alues of p. P"')Q '''j l

"v

.. \I''''''''

" 1 Q"")P ,,,) 211+ I . pO:' II\/" II\!" .......

11+1 I I . Jl'

-

21l+

so Ihnt ns f1- CO the left-hand side of (18.11) lends to - L showing that C = - 1. Writing (18.11) in the form

IQ.0')j

1 l',,(/l) = (fl~ l){P,,(u)P nnd noting from (17.7) that Qn(/t) ~ 0 as It -+ co we hnve d

dlt

J'"

li';

Q.(,'I ~ 1'.0') ,"«'---'-'l""H;"p".('"I"'Il"

(18.12)

19. Recurrence Relations fOr the Function Q.. (,t). nccurrcncc relations for the Legendre function of the second kind Cllll be derived from Neumann's formula (18.1) and the corresponding recurrence relations for the Legendre polynomials p .. tll). From the recurrencc relation (l.~.2) nlld Neumann's formula we have

('11

+ I)Q..+I{Jt) + 1IQ"_I{Jt) =

(II+!) JI -1

Now

! (, I,~

.(1] <1< ~ ,'Q.(~) -

If we write the seeond term

011

(n

+ !I

the right ns

';P,M} cl~. It - ..

f,P, .llldl

J-,po(ell'n(~)d~

we see from (15.0) thnt it vflnishes if " ¥- O. Hellcc wc havc 1 )Q"HVt) - (211+1 )pQ"Vi) tIQ._IVi) = 0 (HU) showing that the functions Qntll) for three consccutivc vnlues of 11 satisfy [I relntion of thc same form ns thnt for thc funcLiolts P n(Ji) (equation (U.2) nbove). From !\cunultllt's formula (18.1) we have

(,,+

+

)=Q'( n Ji

'J'

!

1'.(1) d' -1 (p ;)1"

70

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

t

20

and if we integrate by parts on the right hand side we find that Q;(I')

~

}·Ience

!(-'- (1)"\+ !f' P;(,) d'. ~ 1-" 1 +" ~ -Il'-~ )-If' p;.,«)- p;-.(;)

' ( )-Q',,_1"-. ( Q,,+II'

-I

t

,,~

l

=!(211+1)f

I'

( ..

P,,(E~(I~,

-I!' -..-

by virtue of eqlllltion (U.O). The integral on the right is 2Q,,(I') by Neumonll's formula so that, finally, Q~+I(Jl) - Q~-dJl) = (2/1 l)Q,,(}I). (10.2)

+

20. The Usc of Legendre Functions in Potential Theory. In potential theory we have frequently to determine solutions of Lnplnee's equation 17:!.'}1 = 0 which salidy certnin prescribed boundary conditions. If we hnve n problcm in which the natural boundaries are spheres with centre at the origin of coordinates it is Ilnlurnl to employ polar COONlinates T, 0, cp. In cases in whieh there is symmetry nbout the polnr !\Xis, 'P will not depend on cp so we may write tp = ",(r, 0). It then follows from example 1 of Chapter I thnt

+

(A"r" JJ"r-"-I)v" will be n solution of Laplnce's equntioll provided thnt v" is II solution of Legendre's equlItion (17.1). 'l'uking v" to be P ,,(cos 0) C"Q,,(eos 0) we sec that we llIoy write

•

+

~1(r. 0) = ~ (A"r"+IJ"r- tl - l ) P,,(cosO) .-0

•

~(C.T·+D",.-tl-11Q .. (cosO)

(20.1)

.-0

where the quantities A". B ... C,,{n = 0, 1,2, ... ) are atl constants. Now it is obviolls from equation (18.8) that Q,,(eos 0) is infinite when 0 = 0°, and we know on physical grounds

120

LEGENDRE FUNCTiONS

71

thftt in the case of spherical boundaries II' remains finite atong t.he axis 0 = o. l!cnee we mllst lake = DR = 0 for £Ill vnlues of 11 !mel obtain t.he potential fUllction

en

•

+ En ,.-n-l) l' ,,(cos 0)

.-,

1/1 = 'E (An Tn

(20.2)

which is valicl liS IOllg as r is neither zero nor infinite, Le. if n :;::;; r :;::;; b where 11 and b flrc finite and nOll-zero. If the region under discussion is the int.erior of II sphere, i.e. if o S; r ::;: 11, then to llvoid 'P becoming: infinite we must lake J),. to be zero to give

'p=

•

.-,~A"T"P,,(cosO).

(20.3)

On the other hnnd if the region beillg considered lies entirely outside this sphere, we must lake 'P =

•

~

.-<>

11" ,.-"-1 P ,,(cos 0).

(20A)

Examples of the lise of the solutions (20.2, :I, .~) in potential theory arc given in Coulson's Ell~clricity (Olivcr &, Boyd, 1(018) §§75-78, 80; n furthcr example is given below. The Legendrc functions of the second kind, Qn(eos 0), which nrc abscnt from problems involving sphcrical boundaries, cntcr into the exprcssions for potcntial functiOIlS npproprinte to the space between two coaxial cones. If 0 < a.. < 0 < p < 7t we must takc a solution of the form (20.1). Suppose, for example, that V' = 0 on 0 = a.., and Ip = Eot"r" all 0 = p then wc Illust have An? n(eos a..) find

+

+ CnQ,,(COS a..) =

0

A"P,,(coS{l) C"Q,,(COs{J) = a..", 11" = D,,= 0, the latter rcsulls following from the fnet that if ot *- p, Q,,(cos a..)P,,(cos P) - P,,(eos a..)Q,,(eos fJ) does not vanish. Solving these equations for A" and e" and inserting the solutions in er\lmtion (20.1) we find that in the space between the two cones

72

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 20

1:

IX r" {Q,,(COS Il)P,,(COS 0) -1' ,,(COS a)Q,,{COS O)} n_O" Q,,(COSIX)PII(COSfJ) P .. (cosa)Q,,{cosfJ) (20.5) To illustrate the usc of the solution (20.1) [lnd of some of the properties of Legendre fUllctions we shull nO\\l COIIsider the problem in which lin insulated conducting sphere of radius n is placed with its centre at the origin of coordinates in llIl electric field whose potential is known to be =

'P

" a"r"l\(cosO) :E

"-,

(20.6)

nnd we wish to determine the force on the sphere. The conditions to be satisfied by the potential functions 'P arc (i) that 'P is n solution of Laplace's equatioll; (ii) that tp hns the form (20.0) for lnrgc values of rj (iii) 'P = 0 on

"= a.

The conditions (i) and (ii) nrc satisfied if we tnke 1j!

=;: (a."r" + "_I

~;I) P n(cos 0)

T

and (iii) is satisfied if we write B n = - cr."a 2n+l . We therefore hnve 1j!

=

~ cr." (Tn _ a2:~I) p n(cos 0). r

,,_I

Thc surfnce density of c111u'ge on the conductor is 0=

,a,) = --:--(-a r r_a ·~n

'"

- - ~ (211+1)an-1cr."P,,(eosO) ·~n

,,_1

und since the force per unit area 011 the conductor is 2J1'02 the resultant forcc 011 the sphcrc is in the 0 = 0 direction, and is of magnitude F =

fo"

2J1'0 2 2J1'a2 sin 0 cos 0 dO

(20.7)

121

LEGENDRE FUNCTIONS

where 1",n denotes the integral (2/1

+ 1 )(2m + 1)

r•

cos 0 sin 0 P",(cos O)P ,,(cos O)dO.

Changing the \'nrinblc of integration to It = cos 0 and lIsing the recurrence rclution (H.2) we find that I",,, = (2m

+ 1 ) { (11 + 1) (t",(p) P n+l(/t}dp + on (/",(/I)fJ.>-l(II)d/l}

find by the orthogonality property (15.8) this reduces to the form 1 m .. = 2('11 1 )0"",,+1 21/(')",,11_1' (20.S) Substituting from (20.8) into (20.7) we find that the Lotnl force all the sphere is

+

P

= :E•

"-,

(Tl

+

+ 1 ):t."oX"+l(/2 +t. n

21. Legendre's Associated Functions. \Vc saw in example 1 of Chapter I that the solution of Laplace's equation in spherical polnr coordinates reduces to the solution of the onlinnry diffcrcntinl equation, 2 de m } (J -/(2)(18 --2/1 - + { 11(11+1)--.. 8=0 (21.1) dp dp l-Jl~ which reduces to Legendre's equution when 111 = O. This equation is known as Le~cndre!'s Associated Equrllion. To soh'c this C
Substituting this expression in the differential cquulioll we find tha.t y sntisfies the equution ([2 Y fly (l-!t~)-1 ~-2(1-1II)!~'-1 (,ll"

(It

+ (n+m)(n-IIl+1)!J=O

und differentiating this equution III times wilh respect to by Leibnitz's theorem we find that

I~'

THE SPECIAL fUNCTIONS OF PHYSICS AND CHEMISTRY § 21

74

,P d ) dOy 1 - 2/ 1 -d +11(11+1) - d =0 { (l-/j~)-1 (/' I' Il"

(21.3)

showing that if rJ-yJdJ'" is n solutioll of l.cgcndrc's equation (17.1) the fUllction defined b)' equation (21.2), is a solution of Legendre's associated equation (21.1). Similarly if we put = ('Il! - l)i"'y in equation (21.1)

e,

e

we find that ~

{Fy dW

dy 1 (II

(l-W)-~- 2(l+m)/t- +(II-m)(lI+m+l)y=O,

(21 ..~)

If now we differentiate cqllltl.ion (17.1) III times with resJled to Ii we obillin the equation

d

)d

oy d' .. -2(1+III)-d +(11-111)('1+111+1) -l-~O { (l-/l'-)-d Jl,Il 1/'· showing that if YV') is n solution of Legendre's C
(p' _ 1 )10 d'"y

(21.5 )

dp'

is n solution of Legendre's nssocinlcd CfJuntioli (21.1). Tnking the two solutions of Legendre's equation to be p .. and Q,,(f/) it follows from (21.5) that thc functions

vl)

P::'(fl) =

(Jl~

-

Q::'(p) = (JL~ -

1 ji'" do'")' "VI). m IIlt

1 )i'" d"'Q,,(p) (I/t'"

(2UJ)

arc solutions of Legendre's associnted equntion. As n consequence of equnlion (~1.3) we see that so also nrc the functions

P;;"(,u)=VI' - 1)4'" ( and

Q;;"'(p) = (,,' -1)-1'

J: ... s:

p ,,(~)(~).. (21.7)

r•J•·... J.•

Q.(;)(d.)". (21.8)

121

LEGENDRE FUNCTIONS

It is
Ull

immediate generalisation of (G.:;) that

,

(1:r",2 1'I(lX,{3;y;x)=

(.I.(P). ,

2/'1{a:+I11,{3+mii'+m;x)

(y)",

P~(J.I)

so that using J\lurphy's form (15.2) for tim],

-/(It'"

(I

"II =

75

we sec that

(-11)",(11+1)", ., ( I-IL) ( 2 )"'Ill,, Q/'llll-Il,11+m+lilll+1;--· • 2

Hence P':;(/J-) as defined in (21.6) may be written in the form P':{jl) =

r(n+m+l)

•~"'m IT( 11

Q

1(ll.-l)I'" XQP, m+1 ·

(

1-P ) 1I1-1I,1I+1I1+1;m+l;-2 . (21.9)

Other CXpl'CSSiOllS for the first of the two fUllctions (~1.6) can be easily derived. If we make usc of the result (7..Q we sec that P;:{/t) =

rtll+m+ 1) 2"111,'T( 1/

1/1 +) 1

(/1+1)"-11"'(/1-1)11'" X

, (

Q}'\

-

U-I)

Ill-II, -II; 111+1;'--

ll+ 1

(21.10) find similarly, if we mnkc lise of rc1nlion (7.6) we may derive the Cx!wcssion P;:(jt) = P{Il+m+l) -mlF(1I 111+1) 11 1

(/1-1)1' +

}' (11+1,

X,

2 1

-/I'

,

I-Il) 111+ ,..:.! -. (21.11)

From Hodrigllc.<;' formula (15,3) wc dcri,'c thc simplc cxprcssion 1 dm +" P';;(/I)=;:;n-I(p2-I)lmd m+"(1/ 2-1)", (21.12) _ 11 It The simplc differentiation

76

THE SPECIAL FUNCTiONS OF PHYSICS AND CHEMISTRY § 21

d'" _"_l_~r= (_1)",r(I1l+II+1+2r) Ir"'-"-l-~' d,t"'P1+2r) mn)' (because of the duplication formllln) bc written in the form

r('n+

dm r( Jl+l I- (IJt'"

I(."+tl,(*,,+l),} -. /t"+I+- r

_ (_ )m r(

-

1

III

+.11 + 1 ) ltm+'n+' I,ltm+'''+ 1),' /tm+"+I+~'

showing thill, by tcrm by term dilTerentintion of the solution (17.7) of Legendre's c
Qm(" I ~ (-1)'" r(~) r(m+ 11+ 1 ) ,t-m-"-l(II~_l 11m X 2"+1.r(II+~)

"

~Jo'l (~'IIl+~J1+~" !m+!II+l: 11+%: ;~) (~1.1a) of Legendre's associated equation. Solutions of the type (2'1.7) Ilnd (21.8) ean be derived in a similar fashion. Using the result

J'. J' J' (' - 'j' (d<)m ~ -'- (" I

I'"

1

2

III!

1 1m

"

(m+1).

(' -

2

P)'

in equation (21.7), with :'Ilurphy's expression (15.2) for P ,,(Jt), we derive the expression I ( -(Jl-1)"'~PI-11,1'+1;III+l:

Ill!

fo,

~

r J: ... J:

'OP)

(21.14)

P ,,(E)(d;)m,

and this yields the solution V;;'''(J-l)=

2.., (1'+')im~FI(_11. 11+ 1; m+ 1: IIIJtl-

I-II). (21.15) 2

The solution provided by Hodrigucs' formula (15.3) CAn obviously be written as

n

LEGENDRE FUNCTIONS

I"

,,' I)-1 p-"'I., ) = Vi· .. V

In

0.

2"1l!

m

d"-M _ _ ('j~_ It.

dp."-'"

V

similar WilY the solution

Q__" ) = (_ )_ .. Vi

1

rmr(u - m+ I) . (j,'_Ij-I2 .....1r(II+!) p"-+I x

~Fl(~lI-!m+!, !Il-!m+l; Il+~i~)

r: .. f:

is dcrh'oo from equation (21.8) nnd the J'(11

(21.16)

+ 1)

f~·

(!Il

(21.17)

~lIlt

+ i;~;; + I )r (d$)-

O( ) (!/I-!m+l)r(!Il-!m+l)r =(-1'" . -1+2. ) J 11-111+1 -1, .....",...

llsed in equlltion (11.7). The four functions P';:(Jl), (~::'(Jl), P;:"'(,I), q;:"'(u) defined by equotions (21.0), 21.13), (21.1.l) find (21.17) respectively nrc therefore solutions of Legendre's associated cqlllltion. They lll'C lWOWIl liS Lc~clldrc's Associated functions. Although the expressions I\bo\'o have beon found by I\ssuming III nnd /I to be integers it is rcndily shown lhat the solutions quoted arc valid cvcn when III and /I are nol. integers. Since Legendre's nssocinted equation is of the second degree it follows that only two of these four fUlletions llre linearly independent. and that the other two may be expressed simply in terms of them. It follows imlllediately from equations (21.11) and (21.15) that if /fI, tl arc integers ~-( ,) = F(I' - m 1) 1'-( ). (0 8) '- .. I r(t,+m+ 1) .. Jl _1.1 Furthermore. if we apply the result (7.6) to the hypergeometric series on the right hand side of equation ('!1.17) we find that

+

Q';;"'(II) = (-I)'" r(! )r(tl- 111+ I) ,,"'-"-1(,,2_ 1 )1'" X .

:!'""'-'F(1I+%)

,!Jo',

(!1Il+~1I+!. !1II+~1I+1; .,,+%;

;J

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 21

78

which, on comparison with cljuation (21.13), revcals thc rclnlion _'" r(1I - III + 1) '" (21. HI) Q.. (/l) = r(fl 'Ill I) Q" (p),

+ +

It is aow a simplc muUcr to provc 1hl\t whcll III :mu II arc intcgers, nnd m is rixed. the polynomials P:'(/l) fonn an orthogonal sequcnce for thc intcf\'al (- 1, I), .\Iaking lISC of the N:Sults (21.18). (21.1~) nnd (21.16) we find that

,

I-, l':'(I')P:'I/I)d,1 ~

F(otl+m+ 1) F(II

II

1 11"-'" d"'+m _ _ (,,2_'1 )" _ _ (,,2_1)"'//,' 111+ 1) 2"+""11! u'! _I d/,"-'" r c/JI"'+'"

and after integrating by parts on the right reduces to

q,,+m+')

r(1I

(-1)"111+1) 211+"'II!,,'1

II -

III

I' " 0 ) _I

v,~-I

timcs the expression

v'

)'df.l "d-+" - - 1-1·

c1,,11+'"

This integrnl is evaluated by thc mc1h<XI uscd lit the end of § 15 and wc find that

I

I

_I

F('II+III+ 1) :! I'';(I')P':-llJ)cl/l= 1'( )(-1)"'--0..... (21.20) II

III

+1

211+1

•

In many physical problems I' = oos 0 so that -1 ~/I s;: 1. It is thell not IIlwnys ooll\"cnienl to have Ii factor of the form (/12 - 1 )1"'. We use instead Fcrrer's Cunellon

~I':'(/t)= (I_/t2 ,lm d"'/\Vl).

(21.21 )

el/l'"

WillI lhis notnliol\ we may wrile (2J.20) in thc form r(1l+m+l) ltV' .·v'JU,II= F{rl 111+1) I_II ,/'''',)7''''1.\"

2

211+1

0

.......

The other fonnulae nrc nmmcndcd similnrlr.

(21.22)

122

"

LEGENDRE FUNCTIONS

22.

Integral Expression for the Associated Legendre

Function.

If we assume. Cauchy's l!lt:orclll in the fotm I)

.)-, fc ../(C) dC ~ I(p) - II. __71

where /(1;) is nn lI11nlytic function of the complex "nrinble I; in n ccrtnin domain R which includes the point I; = I'

nnd where the integral is taken nlong n closed contour C which includes I; = I' lind lies wholly within the domnin Jr, thell by cliffcrcllUnting' both sides of t.he equation r limes with respect 1:0 JI we obtain the result

d'/(,,) (Ipr

=~f 2:u

/CC)

c (I; _/1)....1

,IC.

(20.1)

+"

Substituting 111 for rand (c: - I)" for ICC) in this equation we find thnt, as n result of cquntion (21.1), P';(JI)

=

(11I+")! (.U-l)l-f 2" . Ii!

Tf II.

>

(C:-I)"

c (1;-1' )"+"+1

2=ti

d~

(22.2)

0 we may lake the contour C to be the circle

IC-" 1- I vIi"~ - 1II· Integrating round this contour we obtl\in from equnlion (22.2) the equlltioll 1 f2.~

"

_7t

0

cos

VI+·V(/I~-l)eos(tp-'P)}n.

Sill

=

(II

(IIltp)chp

( )pm() +III 1/1)1 COS sin IIlIp II 1'1

from whieh follows immediately the Fourier expansion

(p

+ V(P' -1) 00' (9'- 9'))" 111 )ll~:"{JI)cosm('P-9')' ",_1 '1+'" .

= P,,{JI)+2i (

(22.3)

I) Eo G. I'hillil)S, "'wlt/ioM of a Compute Variable, (Oliver,," lloyd, HHO,) p. 0:1.

80

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 22

Chunging 11 to - (II VI'

+ 1) we

obtnin the expansion

+ ·V(/I''!. -

1) cos .,,}-"-I " (11- /Il)1 = P .(p') 2 1: (-I)'" I ' P':(p') cos m~. (22.4)

+

_I

II.

Appl)1ing Purse,·nl's theorem for Fourier scries to thc series (22.3) and ('!:!A) wc find that the scries ~ (II-m)!... '" P.{fl) P .<1/)+2"",(-1)"'( )1 p .. {fdP.. (/1')eos mrp .. _I

II

+

III.

c..'onvcrges 10 the sum

-'-f' U,+ v'(l"-l)OO'(~+P))"d' (,l + V('j'~ cos 2."1: _'"

1)

lJI}"H

lJI·

This integral may be evaluated by means of Cauchy's theorcm,l) Its "lillie is fOllnd to he p .(1'1,'

+ ,1[(/,' -

1)(1'" - 1)] co, p)} Writing Jt = cos 0, It' = cos 0' nnd cos e = cos 0 cos 0' sin 0 sin 0' cos rp we obtnin the result

+

p .. (cos8)= p .. (cosO)J~ ..(c.:osO')

+ 2 .._I ~ (-1)'" (11-111 )! J~:(c.:os O)P;,"(cos 0') cos (mrp) (11+1II)!

(22.5)

which is orten of value in the solution of problems in wave mcchnnies.

23. Surface Spherical Harmonics. From the two scts of orthogonal functions r:'{cos 0), cos (11111') we ellll (ofm n Ulird set of functions X

(0 ) ~ ...... ,rp

(211+ 1)11 (11+m)!" (II-m)!}io 1·... (cos 0) cos ~:t (111::;,1)

I) !"or delnil$ see E. W. lIobllOll. Op. eit. pp. 805-71.

11/

9' (23.1)

123

81

LEGENDRE fUNCTIONS

which is 1111 orthogonnl set of functions 011 the unit sphere, i.e. I he functions of the set satisfy the normnlizution relation "

2...

sinOc/0J -Y R ",X"' , .... dfp=6",,·"......·. J0 0 ' In

Ii

y

similar WilY we 11 ....

(0. Ijl)=

('CUI

construct

II

st.:l

)! '1'"•(cos 0) . ( ~"+ ')1 {("-"')I}I -0-

(

_."'l

(23.2)

Sill

11+ III •

mrp

(III ::;;;: II)

(28.3)

6 ....· 6",,,,,

(:!3.4)

which sntisfics the rcilitions

Josin 0 f/O Jl!. I' ..... If", .... tiff = e>

0"

"

Jo

2,.. .

sinOclOJ X" ... Y"..... drp=O

(23.5)

0"

for nil integral values of II, n', til Ilnd Ill' with '" s;: II, m' :s;; II'. DeclIlIse of these orthogonality rclnliollships we ciln cslnblish nn expansion theorem which is II slraightrorwnrd gcncrali7.ation of the Legendre series (16.3). It is rcndilr shown lhat for II huge class of fUllctions /. Ull~ fUllction 1(0, 'P) cun be represented by the series

•

• •

,,-0

,,_I ... _1

}".:c"P,,(cosO)+ E

}".:{x"",X", ..(O.rp)

+ V"",Y",,,,(O,rp)} (23.6)

where the coecricicnL.. c", x"",. V" .. lire gi"clI by the expressions

.>//+ I J~'lip e" = ---.I:t

0

=

V" .. =

(23.i)

0

f,sin 0 dO J, X".",(O, p)f(O, rp)drp,

(23.8)

f"osin

(23.9)

.~

X" ...

f' /(0, rpjP ,,(cos 0) sin 0 dO, 2.~

OliO

J2.\-•• (0. rp)J(O, rp)tlrr· o·

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 23

82

For finy gi"en fUllclion/(O. p) the series (23.G) CAll therefore in principle be computed by a series of simple intcgmtions. 'I'he functions X .... and Y ..... which nre known ns surface spherical harmonics can be constnlcled ensil)' from the known e:'<prcssions for the nssocintcd functiOllS T':.

We find, for instnnce, that , ..\ 1,1(0, (p) = -

:l I Car,)

sin 0 cos p.

15)1 sin 0 cos 0 cos p, .\':.1(0, If)= - ( 47z:

-"2,!(0,9') =

15)1 . ( -167t SlIl'OCOS'1.rp,

-"3.1(0,9') = -

G!2~)i sin 0 (5 cos! 0 -

X 3 ,,.(O, rp) =

(105)1 sin! 0 COS 0 cos 2tp. 16n:

X 3 3(0,9') •

= -

1) cos !P,

(~)I sin' 0 cos 3p. 32:r

lind the corresponding expressions for the Y II II' nre ob· tained by rcplncing cos (1119') by sin (IIUp) in Lhe c'xprcssions for the X" ",' The functions X" m nnd Y" '" have the important property that they arc solutions of t.he ]lartial differential equation 1

a (. 0 ax) 1 0' x ,~ an +~o-a--+II("+I).\'=O (:!:J.1O)

~oao sm

Sin

Slll~

p-

so that thc function

(A,"

+ Br"-

l

)X....(0.9')

+ (C,,, + IJr.. -1j Y ....(O, 9').

where A. Il, C and JJ are constants, is a solution of Laplaec's equation. It follows immediately from equations (23.1)(23.9) takcn with the cxpnnsioll (2:J.6) that thc fUllction

LEGENDRE FUNCTIONS

I"

. (')" +:E. . (')"

Ip(r, 0, rp) = E e,. ..-0

P,.(cos 0)

(I

:E -

tI

II_I ",_I

+

{.:r... .Y .,,,,(0. If) + Y..... Y .....(D.Ip)}

satisfies Lnplnces efIulllion ill the region 0 ~ T ~ a, is finite at r = 0, and takes the value /(0,9') on the sphere r = fl.

For example, suppose we wish to find the solution of Laplacc's equation which lnkc.'l 011 the value :c2 on the surface of the spltelc r = a. Here we ha\'c a~ ll~ /(0,11')= 112sin20cos 2tp ="3 -"3

=

(a COS'O-l) 2 + ~1I2sin20cos ':!.rp

,,2 (12 ('\.it\ I 3" -"3 /J2 (COS 0) + 15J ...'(2,2(0, lp)l/2.

Thus the required solution is "'(T, 0, m) = T 'r

2 2.-3 11 2 _ 2.. m) 3 r P,(cos 0) + (2.)1 1:; r2X....(0, 'r ~.M

Substituting the YlllllCS of P .. nnd X .. .. And Ir:lnsforming back to cartesian coordinates we sc·c· that the required solution is

24.

Usc of Associated

Functions in Wave of Ilssocintcd Legcndre functions in wnvc mechanics, wc shnll consider one of the simplest problems in thnt subject - that of soh-iug SchrOdinger's equntion ~ 8.,,2 m J7-IjI+~(IV- 1')'1'=0 (2·U) Lc~cndrc

l\.lcchnnlcs. To iIIustratc t.he

IlSC

for the rotntor with free nxis, thnt is for n pnrticle moving 011 t.he surfncc of u sphere_ Tn equution (2·U), \I' represents the totnl cncrgy of the syslem, V thc potential cucrg)'. In thc easc 'Indcr collsidcrnl iOIl V is Il l:ollslnnt, Vo

84

THE SPECIAL fUNCTIONS Of PHYSICS AND CHEMISTRY § 24

say, and tJle wavc function 'P will be II function of 0, 11 only. rr the radius of the sphere is dcnoted by a thcn equation (2'k 1) is of the fOl'llL 1

ClIp

cotO

a" ao"+ 7

alj1

1

I12p

ao + 112 sin 0 a,,: +

8:z 2m( 11'- Vo) II" '1'=0. (2.L2)

If we consider solutions of the form

'1'=

et'*''''''

thcn

e' 0 + 81t2ml1~( /"II' e" +'cot ,Substituting (:N.3)

I' = cos 0,

wc find lhnt this cqunlion reduces to J~cgclldrc's associntcd equation (21.1) and hence has solution

e=

II P:(cos 0)

+ BQ:'(cos 0).

Howevcr for the same reason as in the case of potential theory (§ 20 abo,'c) we must take n = O. The solulions of equation (2.L2) will thcrefore be made up of com· binations of solutions of thc form 'Pm ..(0, tp) = A",,, 1::Hm 9' 1'::' (cos 0). where A .... is a constant. Thc physical conditions imposed 011 the wave function 'P nrc that it should be single-valucd and continuous. Obviously thcn thc 'physical' solutions will havc III lIll inl.cgcr, sincc 'P... "(O.!p 2:t) must equnl !jJ... "(0, p). Further in order Uillt the series for P'::(/l) should COil verge for the vnlllC5/1 = ± I it is necessary that it should havc only a finitc number of terms. This is possiblc only if II is a positive integer. If therefore the solution (2.~..q is to be ,'alid for 0 = 0 nncl 0 =:z we must hn,'e " n positin'

+

"

LEGENDRE FUNCTIONS

integer. The phy!>ieul conditions on the wave function arc therefore not satisfi(.'(l by !>)'stellls with IIIl arbitrary value for lhe energy 11' hill only by systems for whieh

h' + -.-,11(/1 + 1). 8...-11I(1

11' = 1'0

(2.1.,5)

where 71 is a positive inleger. Tn other words, the energy of such II mechnnical system does not vary continuously, but is enpnble of nssuming valucs takcn from thc discrete set (2.1.5), 1':XA~II'LI':S

Show tlmt, if" u odd

1.

".to) _

O. ami that, if II (-I )i·

n'

i~

.:....'11.

2·{UIl)!l· 2,

I'ro\'c thut

I:

JI··'

.._en +1 I f ' _ I'

3.

(I)

(ii)

+ "lV"

(1 - "O-~(I

-

P_VI)_iIOg{I+JI} I-p

I) show thut

•

-lltcH - 1: I,·P.V')

.-,

r(II"'U

".VI) - IIlru)C·.F,ll.

-II;

i-no C-')

lkduco: tlmt (lU)

-4.

Ir"

...... U. -II;

i~

f hen~.

l

P .VI)( I

-

mnking \I$e or

I_,l I

-1'--

I) -

(II

+

il

" l)
and

Il!l'\i)

1-'1;

2Jlh

+ II')-i dJI

Hodri/{lI~'

l _1,')_(1 - 211h

"h-

~

-

_II

+

I

rormulu. dellullC l.IUlI

+ h')---j dJI

-

2""(111)'

";:c-''7o (2" + 1)1

86

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

P'~(J:)

_

• I:

(211 - 4r -

l)/>~_"_I(;t:)

~.

where II _ Hn - I) or III - I according us II is odd or CVCIt. Deducc HUlt for nll;t in the closed intcrval (-1, 1) nnd for 1111 positivc intcJ;el"S u, the vnlucs of the fUZlctions II' ~(:I'") I. II-'l P' ~(;r) 1, n-II J'''(;I:) I, ' .. CUll never exceed unit)',

,..

I'rove thnt

I' I'

p.{JI)cf}1

,

-;;--+ I {P~_I{,U)_II

nnd deduce, froni eXflllllllc

I

I,

Ihul if

II

is un odd integer

(-1).~-i{n-1)!

I

o''.VI)d}1 - 2.(in+I)!(ltl

Whot is thc v"lue of the inteJ;nll when 7, Using equation (J.I.2) nml tin; llmt if " is evell

I

I1!.~ults

II

ill is cven?

of the lfIst eXlllllple show ('.-2)!

t

o}IP~(/I)d}1 - (-1)lft-1 2ft (IIl+I)IUIl

1)1

lind that the illtegnll hus the yuille zero if II is odd.

8.

If Il~_

I

-I

11/1

l'"{/l)P~_I{,II)II

+ 1 )lI ft +1 + ,UI.

prove lIml (II

9.

I

_ 2. Ilenec c\'nhmtc ll.,

If m and " atC positive integers, prove that

I

2OH~H{(,tI+Il)l}'

I

-J (1+/l)-"P.(JI)d/l-

10,

III !(III +211

+ I)!

If ,. is eyen nnd III > - I, prove thnl

IO}I'"P~(;I}d}1 I

Deduce thaI.

r(illl+llF(im+l) jn+l)

- 2r(jlll+jll+.})lUm

LEGENDRE FUNCTIONS II.

81

Show that I~.(,,,)

-

("+')" .p, (-II; ~

-II

,

"-')

1'--

'/1+ I

nnd hence thnl

Deduce thnt 1'. (cosh ll)

li: I

12. If 11}1) _ I}l' - I)· 8110W, hy llsillj:: nolle'll lhcorcnl thnt 1'(/1) 11\1l1lllll\\'e at least one u:ro hel\\'cell - 1 lUlil I. l'roceeclillJ: in this wny deduce lhlllp·I(}I) IIiU 11 zeros bcl\\'ccn _1 Hnd 1. IIClIl'C show thnt when" is e\'en tile reros of I'.{/l) occur In I'aln. equlIl ill 1IIl1gnitude but 0llllosite in sign, IIml thlll whell 'I i~ odd, 'I _ 0 is II zero lind the others occur in ellUl,1 "ml 0PIKlSHc plIil'll. 13.

If Y(}l) is nil}' solution ot the linrar dittcrelllial Cfluatioll dly

.C!<) -fl}l'

dy

+ pCp) 11}1 ~ + r(Ply -

0

in which Qt. P lind r lire continuous tunetions ot}l whose derl\-at'"u of all orden nn': continuous, Ilro,'e tlmt yl.Jl) Cllllliot 11:1\'1': nn}' repented uros except JIO$Sibl}' for ":llul';S of /' which IIlltisf}' tile «JIIlItion a(ll) _ 0, Ikdul'e thnt all the zeros of J>.(lI) nre distinct. 14, Prove that the Legendre jKlt}'nomial p.(z) 11lI.'I Ihe smallest distance in the menu from zero of nil pol}'nominl.'1 ot degr« 'I with leading ooctricient 2"(1).{1I1 15,

If Il denotes the opemtor

!!.{{I-zll.!!.-} ,.., '"

pro,'!': Ilmt

f

l P.(z)R{f{z)}d.z -I

pro,;ded thnt I(z) lIntl Pro,'e Uml if II li: I

f-,

-

rex)

- 1I(1I+I)f

1 _I

p.(zl(z)d.z

nn': finite nt z _

l log (I - z)P.(z)dz-

±

I.

11(11_1)

88

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY Pto\"e til!'t

16.

~\

".(",LI. ,...

I

2

J + ",)"" J J' {,,_;r+ HJ + J

Ii)

z)-~

_I "'(I

2h-

'>.(.r)d..r

I

(ii)

2JU'

-I (I

I -

h'

Ifn'_1_21u:+',', Ilto\'cthllt iflhJ
17.

(i)

_lloS

I-z

l

iil 18.

_I

log (1 - hz

21,·-'

P.(Z)I/.r-(2U+I)(Il+1) 2h~

Il)J'~(x)ILr _ - --;;;;;::;:-~, '1(211+1)

If

e"-' J" "-----':"----="1 + tI~,

1(...., a) -

0(1 - :!~z

(m> I)

~')

pro,"" thnt

2tl.f"

I

J

-I If".. f1)'>.(z)dl" -

19.

If

= it

real nnd

.• J

1=1 <

01

Pulling

=_ :':11"'(,11' -

(II

+ m)(211 + 1)

1 pto\·e thnt

~

+ =OO!ttp -

,

,/(1 :') I){(I -1I}1) where 11 is so smnll thnt

]hV' ± "'()I" - 1) COJlrp} 1 < 1 (0 ;S tp ;$ :I). ClI"l'ntllliUK holll sides in powcrs of " ami equuling coerrieicllts of ,,- show th'll '>.(,11) -

.

~ , J'" (p ±

Vf},1 - 1 1CO~ 'f'}·tltp

Hence C\'alunte thc 'IIrn

,-.

" ·C. ".(005 0) 1:

20, Sho,," by making the lIubstilution :: _ the fomlUla in ex. 1D th:'l

± h v'(}.'-I )/(IIJI -I) in

89

LEGENDRE FUNCTIONS

21. "'nkill~ use of lI,e illlcgrni expressfon fur 1'.(/1) derived in ex. III show thnt

•

:E ·C,(I

p.{Jl) -

-I")1·-1'Jl'1'._.(0)

'-
Deduce thnt 1'~(,lIl

1',(/1)

1'.(,1)

J'.(O)

o

1',(,11)

I',(p)

1',(;1)

o

1'.(0)

o

I',(P)

1',<1')

1',(/.)

J'.(O)

o

1"(0)

Pro\·c Ilmt U J'

22.

>

-(I-II')'

I

II

1

Q.(J') - :!-+,

nnd dedllcc IIml

J:

12.(;1) -

J'.(O)

(P -

(I-I')'"

_I

v'

I)h,l/l

I) cosh 0)·118

,/(;1' -

+

where at - i lOA: {J. 1 )11.1. - 1). Hence fintl cxprc!l!lions for Q.(J') nnd Q,(J')'

23. I)clennine the simJlle eXJlression~ for 12.(/.), 12,(;,), Q.(;I) nud Q,(P) b~' workinA: Ollt Ihe \'nltl~ of t.he JMlIYllorninl 11'._1(;')' oecurrinJl in C(IIH'lion (18.8), for these \':.,lllcs of II. 24.

E.stablish thc following formuloe
,..1'.1.1,)12 • fJo)

I ....(ell'.o~('I di, < ,.; I I --.-di + - -+ , I· I

I

I"'.(;I)Q.(;.) -

~1'

1

In

-I

11

~(I'.(~IP

I

/''''(,I.'-I)Q'.(/.)-j

/I

I)p' (!)

1'-

-I

I ("

;-

-I

Provc llml, ir

2n

~"(';'-

(P'-I)P_(P)Q'.(;')-iI 25.

1

l' - ~

-I

I

(iif)

~

1

II';,

-$

I)p (t)l" II

11>111;

(.)

·;11';,11>111.

Is n positive inlelter lind C_ J'

"I'·{cosO)liinOrIO 12.(;,) - COl 2CCOliO + t'

I

+ v{,,' -

I),

90

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

Deduce thllt, if

It I >

Q.(P)-

I,

1: t-.. f" J>.(oosO) ain mOdO.

",_I

0

Uy cvahutiug thi, inteJ,:ral sho..' thnt

\1;01

Q,·,)_---C-·-I P(I .\1 r(lI+t) "" 26.

Pro\'c tlml, ir

III

is n po5lll\'C (nleltCr,

..

1: 1,·-" 7'''(/l) -

r1_'" 27.

"+I"'+t·C-') , .

•

Show IILut. if III I > I,

(_I)"'(2".)I(I_/l")i"'

2-ml(I_~/l"+lllj"'ti

f1> - I then

.. l'(n+m+ll (Pl-Ili-fl (I-P)·d; Q.(P)- 1'("+1) ~,- -I (P (')..... , Deduce thnt. if III

.. Q.(P)-

+ I I>

2 then

V:lno+m+11 vl"-I)i· ( , ) 2.·'r(Il+0 lfl+IJ•••• 11".. (1I+I,F1+m+I;211+2;JI+l

Find n llimille exprusion for Q:4'(p).

28.

Pro\'e the fol1owlnJ: recurrence relations for Ferrer'a t\nocinlcd Legcndre Functlons:-

(iiI (iii)

(0-11I+1)7':.. 01) -

(211+lj/l'l':01) + (11+11I)1':_1(,11) _ 0;

r,:"'(ll) - 1:::(p) _ (ll_m)(t_,,'ji T =(p).

De.ri'·e thc exprnsiol1ll for 1';"(0,1') nnd .\';"(0, '1') for m _I. 2, 3, ,~, ,ill 0 cos" 0 em 'I' in teTIllS of lurfllce Illherical hmnolilcs. 29.

Express the l'lInctions sin"O Iln" IF. lin' 0 01)51 fI"

30. Find the runction which interior or the III here z' y" IIlld tnk~ the \ouluc oz.rl fly'

5Rthrl~

l.nlll:lce·, equnlion In Ihe

+ + ;1 _ f1" remnirl!l finite lit thc origin + + y:" 011 the lurfnee of the .pheft'.

CIIAI"'1o:1I IV

BESSEL FUNCTIONS 25. The Origin of Bessel FUllc(ions. Bessel functions were first introduced by Bessel, ill 18::?l, in the discussion of n problem ill dynnmienl nstronomy. which mny be described ns follows. If P is n planet moving in nn ellipse whose foclls S in the sun and whose centre nnd mnjor axis nrc C nnd A'A respectively (cr. Fig. 8), then the angle

Q

c

S

A

Fig. 8

ASP is cnUcd thc trite ""omtl1y of the plnnet. It is found thnt. in nstronomical cnlculntions, the tme nnomnly is not n vcry convenient angle with which to denl. Inslend we use the IIICOIl ol/ollloly, C. whieh L~ defined to be 2n: times the mHo of the nrca of the elliptic seclor ASP to the IIren of t.he ellipse. Another nngle of significance is the eccentric llIlOlI/tlly, Il, of the pin net defined to be the nngle ACQ where Q is the point ill which the ordinate through p met:L~ the nuxilinry circle of I he ellipse. It is rendily shown l ) by n simple gcometricnl argument I) Cf: D. K HUlherfonl, C/lutiool.ltuhania, (Olin!r« Uo~'d, 1051 )f.1::l.

"

92

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 25

that, if e is the ~eenb'ieity of the ellipse, the relation between the mco.n anomaly and the eccentric anomaly is C=fI-e.sinu. (25.1) The problem seL by Dcssel was that of expressing the difference bctwecn the meflll nnd eccentric tlllomnlics, u· ns a series of sines of lmlltiplcs of tIle Olelln nllolllnly, i.e. thnt of determining the coefficicnts ",(r = I, 2, :J, ... ) stich thnt

e,

•

I; = Z c, sin (rC).

(25.2)

-,

1/ -

To obtain the \"alues of the cocrficienls c, we multiply both sides of equation (25.2) by sin (.e> and integrnte with respect to C from 0 to =to We then obtain f"(t£ o

C) sin (8C)(11; =

I: c, f"sin (rC) sin (se)(lC 0

,_I

Now

!:;rJ,.•

J:5ill (rC) sin (sC)dC =

fmc! nn intCf:,rrntioll by pnrls shows that

I:(U - C) sin (.rC)dC =

~ [,,- II) cos (.rC)]: + ~

r: (~; - I)

cos (.reldC.

From (25.2), C- II is zero when t; = 0 and when t; =:'1:, so that the 5qunre bracket vllllishcs: the integral enn be written in the form

, f",cos al; dl£

-I

.f

ami hence, using equntion (25.1) we obtain the rcsult c.

=...:.... "cos {'(II :r.r

0

e sin lI)}dlt.

(2ij.3)

The intcgrnl on the right-hand side of equation (25.3) is

12S

BESSEL FUNCTIONS

93

n function of , and of the eccentricity e of the plnnet's orbit. I r we write

J .. (x) =

..!.. roos{z sin 0 :t J 0

110),10

(:!5.,~)

it follows from cqullliollS (25.3) and (25.2) that , ~ ( )sin (rC) u-r,=2,L,J.cr . r

(25.!:i)

._1

The funclion .1,,(3:) so defined is called Bessel's cocfrldent or order n. We shall now show that J .(x) is equal 10 the codficicnl of t" in the expansion of cxp Ox(t - ,.-1 in other words we may define J ,,(x) by means of the c"xpnnsion

no

cx p {!z(t_2-)}= I: '/,,(:z:W. I ,, __
(25.0)

To prove lhis we need only show thnt the,f ,,(x) of (25.0) cun be expressed in the form (25A). We first of nil obscn'c that

•l:

•

11 _ _ "

(1)". ~ l: ".(x)/'

(-l)'''_.(x)t"~Il _l: ".(x) - _ 1lO I

n __ OD

sincc both c..'Xpnnsions nrc equal to cxp Hx(t - III}). Equaling' coefficients of til we hn\"c ( - 1)".1_,,(3:) = .1 .. (%). (25.7) In the expansion (25.0) we mny write t = el ' to obtnin the relation

•

exp(ixsinO)= 1: .J.,(m)c l '" l\laking use of the result (25.i) we see thnt Ihe series the right enn be Pllt into the form Jo(z)

• + 2..._1 ~ J:~(z) eos (2/110) +

011

•

2i!: J:",+I(z) sin (2111+ 1)0 ... _1

.

so thnt by equating renl nnd imnginnry pnrls we obtnin the expansions

_.

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY t 26

94

+ ~:£• J,.. (.r) cos (2mO). • 2 I: J!"'+I(.:r) sin (2m + 1 )0.

cos (.rsin 0) = Ja(.r) sin (.rsin 0) =

....

(25.8)

(:!5.9)

If .....e 110W multiply (25.8) by (."()§ uO. (25.9) by sin ,,0, integrate with respect to 0 from 0 to :'I, and usc the formulae

~ .. I"ocos (1110) cos (110)(10 = I"sin (II/Olsin (110) dO = ::...-()"', 0

2

we obtnin the fOl"lllullle

, I",oos(zsinO)cos(1l0}tJO. -.!.. , I"Sill , (oX sin O)sin (nO)dO,

J ..(x)=1

(11 e\·en).

(:15.10)

J ..(x) =

(n odd).

(:.l5.11)

Ucclluse of the periodic propcrtie. of the trigonometric functions wc know that the inlcgr:\1 on the right of C
-.!.. ~

I",{cos (oX sin 0) oos (110) + sin (x sin 0) sin (IIO)}(/O

which is iclenticnl with the expression (25 ..q. I II pllTl.iculo.r Jo(.r)>=..!.- r"cos (.rsin OldO JI

J0

(2.'U2)

In whnl. follows we shalll\Ssutne thnl the nessel functions of Ule first kind nre defined b~' equntion (21U'I) or, whieh is equiynlent. by equation (25A). 26.

Recurrence Relations for the Bessel coefficients.

1f we differentiate thc gCllernting equation (25.6) with

respect to z obtain the relation

;_ (I _ 2.)t cxp I,r(' - -'t-)} - .. __E .I;(x)I" 00

"

BESSEL FUNCTIONS

126

.

.

which is cquivn lcnt to

., -

:E {J ,,(or) ,n+!

-

,

1: J ,,(x) til

J ,,(X)t"- I} -

=

O.

,,--..,

11 _ _ 00

Equnti ng to zero the coeffic ient of ttl we oblnin the relation (26.1) 2J~(x) = J II_I(Z) - J ,,+1(.%)' On the other hand, if we differe ntiate (25.0) with respect to t the rcsultin g equatio n is

~ !X(l + r:)cxp{!z(t - ~)}= " __ t

OIl

".I .. (Z)I"-1

lind this is cqlllvn lcnl to the relution

• !.:r :E on + l"-~)J,,(z) " __ CIO

•

:E 1I.1,,(x)I"-1 = O.

11 _ _ '10

Equati ng the coeffic ient of t"-I to zero we obtain the recurre nce rein lion 211 - J ,,(x) = J 11-1(.1:)

+ J ,,+I (x).

(20.2)

11.1 ,,(z)

(26.3)

z Adding equatio ns (20,1) nnd (26.2) we find that .:I".I:(x) = xJ II_I (x) -

nnd subtrnc ting equatio n (26.1) from (26.2) we oblnin a:.I:(x)

= n.J,,(x) -

.TJ,,+I(x)

(2GA)

PUlling /I = 0 in this last equatio n we have the import nnt specinl case (20.5) J~(.l:) = - J1(x), 1 in equatio n (20.3) we find that and putting

,,=

1 , J)(z) =Jo(z )- -J)(z) .

z

(20.G)

Differe ntiatin g both sides of (26.5) with respec t to x find

96

THE SPECIAL fUNCTIONS Of PHYSICS AND CHEMISTRY § 26

making use of the rcsult (20.0) we havc J o" (.1:) = - J o(.1:)

+ -1 J (.1:)

•

1

which, us a consequcncc of equation (20.0), may be written "

J o (x)

1, + -Jo(z) + Jo(z) =

(20.7)

0

•

showing that y = Jo(lt) is a solution of the differential equation lPy 1 dy+ (20.8) y=O. 2

-+-dz the IV

We can show similarly that the Dessel function J .. (It) satisfies the differential e<Juntion

rPy

~

(." + -;1 tly dz + 1 - zi) Y = 0

.

(II, mtegral).

(20.9)

For, from equation (20.4) we find, as n rcsult of diffcrcn· tiating both sides WiUI respect to x, that

g:J:' (x) + J:(x) = nJ:(x) -

J oo+l(Z) -

xJ:+1(x}.

Now, from clluation (20,4 l,

,

,,2

.-

"J .. (x) = --.; J .. (x) - I1J ..+I (It:),

and putting that

tl

+ 1 in place of tl in cqulllion

(20.3) wc see

so that "

:cJ.. (x)

+ J .(z) = - J .. (.:a:) • ,

'It

xJ..(z)

which shows that J .(z) is n solution of equation (20.9) provided, of course, that 11 is an intcgcr. As we pointed out in § I, cquation (20.9) is known lIS

•

§27

97

BESSEL FUNCTIONS

Bessel's equation. What we have shown is that if the II which occurs in llc.'iscl's c, for liS we SltW in § 1, tile cquulion (2n.fl) arises naturally in boundnry value problems in mathematical physics. 27. Series Expansion fOr the Bessel Coefficient. We slml now find the power series expansion for the Bessel coefficient J n(xl. If we write exp {!x

(t - +)} =

cxp

C~xt) cxp (- ~)

and make usc of the power series for the exponential function we obtain the expansion

.I(xl-i

cxp

')I_~(XI)'~(-X)' -r-ozrrl::.o2'I's! '"'

•

""

a: r+,/'_' ~EE(-l)'(-) - . .-0.-0

2

tIs!

(27.1)

By our definition (25.6), the Bessel eocfficicnl J ,,(x) is the cocffieicllt of t" ill this cxpansion. If '11 is zcro or 1\ positive integer, wc find that J ,,(x) =

~ (_ I)' X "+2' t _0 8.(1/.+ s). _

1!

(0)

(27.2)

and whcn It is a ncgnti,'c intcgcr wc call deducc the scrics for J,,(.v) from cquntion (25.7). Writing' cquation (27.2) in thc form J (x) " we see lhat

x" ~ 1 ., l, ( - tx-)' 2"f1I._os!{n+ 1).

-

x"

.J ,,(x) = 2"nl oFdll

+ 1;

-

{X 2).

(27.8)

98

THE SPECIAL fUNCTIONS Of PHYSICS AND CHEMISTRY § 27

The nl.riution of the Bessel coefficicnts .Jo(x), J I (x), x~ 20 is shown grllphicnlly in )"ig. 0. Thesc urc thc Bcsscl cocfficicnt.s which occur most frc-

J 2 (x) for o:s;:

-0·:)

'-"~~~~~~~~~~~~~~-'-'

Fig. 0

Vnrintion of J.(x). ",{al) and .'.(z) with r.

quclltly in physiclil problcms lind thcir bchaviour is similar to that of the general coefficient ./ n{;r}. Simple relations for the Bessel coefficicnts nul.Y be dcrived ensily from thc series expansion (27.2). For exnmplc, since this equation is cqui\·alent to III (_ 1 )...r2,,+2, ( 1 ) "+2' '2

IX"J (1:) = L

,,'

._0 a!(n+ a)!

(27.·Q

it follows, as a result of rliffercntialing bolh sides of this equntioll with respect to x. nnd making usc of the fnet that (211 2a)/('I1 a)! = 2/(n - 1 a) I, that

+

+

+

121

"

BESSEL FUNCTIONS

d

~

dx{X".J..

(-

(.:r)}=,~osl(1I

I 11-1+:'

l)'~""':'-I

1+9)!(2)

which. by cOlllpnrison wilh (27"\), shows LilnL d

,.

-, {x"J .(x)} = x"J __ 1 (X).

(21.5)

.,.

If we wriLe this result in the form " (x"Jn(x)} = :I:"-I.I,,_.(X) -1 -/

we !'iCC tlllll if

//I

is

fl

positi\'c integer less thun 1., thell

d (x1 dxl x"J,,(x) =

Z"-MJ .._ ..{X).

(27.6)

Similnrly we Clln estnblish thal

d - {....oJ (.)) fix" ur, which is the ( II

SfllllC

~

_ .-".1

.. ~ I

(.,)

(27.7)

I.hillg,

x1 (Ixd) {.:rIOJ ,,(z)} =

- ,r-n-IJ II+I(X),

result which may he gcncrnliscd

d)' {x-°.l .(xl) ~ (-

1 ( -x -/ (J

10

the form

1)-.-"--.1 "+.(x).

In pnrliculnr we have the relntioll z-"./ .(x)

= (- 1)" ( XI dxd)"Jo(x).

(27.8)

which shows how the Dessel cocfficicnb ./ ,,(x) llIay be dcri\'e<1 from Jo(z).

J\nothcr interesting properLy of the Bessel coefficients . also follows f!'Om t.he power series cxpulLsioli (27.:J). This conccrns lllcir behaviour for small \'aluc,.. of t he argument a:. Sincc lim OP1(II 1. - iz2) '""" 1 .-.0

+

100 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 27

it follows from equation (21.8) thnt lim x-"J ,,(x) =

_0

~I. :.!"11

(27.9)

Tn other words, for small vnlllcs of x, the Bessel coefficient J,,(x) behavcs like x"/2",,1. 28. Integral Expressions for tho Des scI CoeUlcients. We have already derh'ed one integral expression for the llessel coefficient of order 11 (Cflllfllion (25.4) above). Tn this section we sholl consider other simple integrnl ex· pressions for these coefficients. We shall consider the integral I ~

,

I-,(1 -

(S)"-le''''cU

in whieh '1 > - i. If we develop exp (ixt) in asccnding powers of ixt we sec that the vnllle of t.his integral is ... (ix)' L -

I' (1 -

f 2 )"-lt·(lt.

$1 _I If 8 is an odd integer then the corresponding integral occurring in this series is zero, and if 8 is an even integer, 2r, say, then the integral has the vnlue • -0

(I _ I o so tbat I

+

+

11)"-1,,'-1 dll = r(1I !)r(r !) r(r1 r 1)

+ +

l~ E(-I)'x'l1"+!('+.1

+ r + 1) (_ l)'x 2' P(! )r(ll + !) !o rlT(1I + r + 1 )2 .-0

(2r)!

(" ...

=

2'

sincc, by the duplication formula for the gamma function, r(!)(2r)1 = 2~'r!T(r+ !). It follows immediately from the series expansion (2;.2) for J ,,(x) that (Iz)" J "z ( ) -- T(i)F(lI+!)

I' (1 -1

<1"--1 "'dt

-1-

e

( 28.1 )

BESSEL fUNCTIONS

!29

101

and it is easily shown thnt this is equimlenL to the formula

."

J .(x) = 2-- 1F(!)r(1I

+ I) I'0 (I -

In pnrliculnr J (x) =

II

."

II'

r-)"-I cos (xt)dt.

(28.2)

2 cos (.Tt) dt (28.:J) o ':'t 0,\/(1 r-) The result (28.2) Illay be expressed in n slightly different form by means of l\ simple change of variable. If we put t = cos 0 we obtain the integral expression J .. (x) = 2" lrH)r(lI+~) 0 cos(xcosO)sinznOllO, (28A)

while if we make the substitution t = sin 0 we get the formula

."

II.

J,,(z) = 2" 'F(!)r(Il+!) 0 cos (xsin O)cos'''OdO. (28.5)

The pnrticulnr forms appropriate to

/I

= 0 nrc

Jo(x) = 2 II"cos (x cos O)llO = 2 It"'COS(XSinO)dO. (28.0) :00

1(0

29. The Addilion Formula for Ihe Bessel Coefficients. In certain physical problems we h1l\'c to reduce n Bessel coefficient of type J ,,(x y) to a form more nmelll\ble to computl\lioll. Wc shall no\\' dcrivc IIll addition formula which is of great usc in I hcsc circumstances_ From the definition (25.0) we ha,-c thc expansion

+

ex p {

HI: + y) (t - ~)} =

J..

J,,(X

+ y)t".

Writiug the Icft-hand side us n product ex!' {

!.(,- +)}. ex!,{ !"('- +)}

lind inscrting the npproprintc series from (25.0) wc find that

102 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 30

•>: ",,(.11-1-

•

y)I," =:E

•1:

".(.'1:)".(y)l,r+.

Equating eoefficicnts of '" wc obtain thc addition formula

•

J,,(x-l-Y)= :E ".(.11).J,,_.(y)

(29.1 )

~-.

'1'0 put this in a form which invoh'cs only Bessel coefficients of positive order wc write the right hand sidc in thc form _1

"

m

1: J.(X)./K_'(Y) -I-:E J.(x)J .. _.(y) -I.... 0

r_a:>

>:

.Ir(x).I,,_.(y)

._,,+1

and notc that bceause of thc relation (25.7) thc first tcrm can bc writtcn as

.

-,

l: (- I )'J _,(.)J ._,(Y) = l: (-

I)'J ,(.)J o+«Y).

r_1

...._
Similarly the thin] term is equal to ._1

r-l

•

•

:E .I,,+r(a::)J_r(Y) =:E (-l)'.1,,+.(.11)J.(y)

so t1mt finnlly wc havc J .,(1l1 -I- Y)

= >:• .I.(a:)J ,,_.(y) ,-0

_.•

-I- :E (-1 )'{J.(x)J .,+.(y) -I- J lI+r(x)J r(y)}. (29.2) 30. Bessel's Differential Equation. Wc showed previollsly (§ 26 above) that, if n is an intcgcr, J ,,(.1:) is n solution of nessel's equlltion (20.9). We shnllnow examinc thc solutions of that C{j1l1ltiOll when the pllrnmeter n is not neeessllrily un integcr. '1'0 emphasise that this purnmetcr is. in gcncrnl, non-intcgrnl, wc shall rcplace it by thc symbol I', so that we now consider the solutions of the sccond ordcr lincar diffcrcntial cquation cP-y

l/:~

+!.x rly -I- (I _ ":) Y = O. dlt x·

(30.1)

-! _) '( t ( 1


SU lId:'jul

3d"1 JO \l0!11110S "!S1Iq ;nll lUln ;x)S ..

U1JOJ ;:1111 U! 111d

Z ... (; -.IZ)·q;(;;

d.\\

+ :t)i-/ =

'~3

0,7

"q

"UUI

'l"!!j.\\

llOISS:lJdxa uu

+ '1(:;) ... (;_.1;; + .1(;)( ,1 0 + .1(;)

"--'-''--''''''"''-''-''=-';''':/.7(:-,--'-')'--''''-''--'''-''.:~~ o=

=

.-

~3 puu

i+"~j

;):'Illl lStllll .1,\\ '(; <: J lIU JOJ lKIUS!lUS df[ .{UUI (1"m:) lUlj1 Ja!'JO Ii! ';)all::n! puc OJ;)Z aq 01 I;) <:IliU1 ;)JOpJ,H[11Sntll ;),\\

(1"m:) ,(" .. 'U '0 =.1)

'·-'3 -

=

(z,' - .(,1

o = hoi - d 1 + It)}

+ . . )}

J;)

IUJau;J::! ll! pllU 1;1

;)Autl lSIIUl ::l.\\ a:lll;)H

0-'

'0 =;;+.+-,.1:-"

3:

•

0-'

+ .+.>:J]J.1bt- (.I + ,I) +(l-·I + ,I )(.1 +.I)} 3: •

lUl[1

aq lsnUl

1i0llS

J;)

SIU;)P!JP0;'

;)In lUlU ;);)S ;).\\ ((;'OS) uonunb" U! S:l]J;)S SPI1 ::lll!1nmsqnS ,.,

'.+.x J ;) 3:

(c'm:)

•

= fi

\UJOJ ;nll JO S! \lonnlOS lS-l!J ;)111 U;H!.f, 'J;>::lalU! uu lUlIl asoddns nuqs :).\\ lIu JO lSJ!~[ ',1 1= = a 5100.1 sull S!111 puu

JOli OJ;)Z J;)lll!;)U S! "

0=;;<1-;;0 ~.10P.1;:HIl S! (~.\Oqu (~,..C) 'p) tlO!IUnU;J IUP!ptq ;JllJ. '~,I - = °b puu l = °d 'f; § )0 1I0!lUl0tl .HIl U! 1Ul{1 pUll lll!od .1U[n1fu!s Ju[nfl<J.1 U S! 0 = x lll!od <JIll lUI{1 ;J;JS ;J,\\

(;'0(;)

'0 = fi(.x ~

xll .xp + .,1 - ) + -IIp x + ~~:t' ~

h;1J~

1U.10J <J41 U! tlo!1unb;J vl{l 1fU!l!J.-\\

'"

5NOIDNn~

0<1

135539

104 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 30

.,.

~

(- jx')'

Y=2'1'(11+1),:-orJ(v+ I),

(30.5)

Comparing this scrics with thc scrics (27.2) wc sec that it is of precisely the slime form as lllllt equation, the only difference being that tl is replaced here by II. If we take the scries (27.2) to defille the Uessd fUlletion of the first kind of order 11, even when II is not all integer, then we lUay write the solution (30.5) in the form y=J,(Il:).

Similnrly, if we substitute a scrics of type

to corrcspond to lhe second root of the indieial equation, we find that it must be of the type

y=

",'

~

(- jx')'

l (30.6) 2'Q l'+l),~rl( 11+1), and with the extension of the definition (27.2) to lionintegral vulues of v we may write this solution in the form y= .I_,(x). Thus when v is '1I0t nn integer we mllY write the general solution of equlltion (30.1) iu the form y ~ AJ,(x) BJ_(.x) (30.7) where J,(x) is defined by the equation

+

x'

J,(x) = 2'r{I'+ l)OJ,\(v+ 1;

-t.1:2).

(30.8)

It should be observed that the results of § 27, 28, with the exception of (27.R) arc truc whcn tl is not nn integcr, since they were derived dirceLly from the definition (27.2), which is equivnlellt to (30.S). The I.rausition is effected merely by replacing" I by r(v + 1). When l' is zero or 1111 illtCgCI' we know from equation (25.7) that the solul.ions .I,(x) and J_.(.-r:) nrc not lincarly

po

lOS

8ESSEl FUNCTIONS

independenl. We must therefore usc the formulae (3.8) to calculate the second solution. We shall consider firsllhe effie in which v = 0. Ii we let

•

....

lD=~Cr~

then in order to sllliify the recurrence relation (30..~) we Illllst have IV

and putting

(!

=

(l

';. t(- txt) ._oT·(e+ I

(30.0)

-.1:" ...

r

we ohillill the first solution (30.10)

tl'o = JoVr).

Using the result

a

of!

I

(I}

+ 1 >. =

I

-

(Q

+ I),

{~_

1

.~t Q + 8

}

we see that

a" ~ -=uJlog.r-~l.

of!

(-tx)' {~ .... -I - } .

...... rI(!? + 1)• •_1 e + 8

PUlling (! = 0 :lnd substituting the \'llluc (30.10) for Wo we rind that the second solution (Orc/oQ),...o is ~

._1

Yo(x)=.lo(w)log,T-l

(-t x )'

(I)~ p(r).

r

(HO.l1)

where

per) = i:.~.

.-1

(30.12)

$

The function Yo(x) so obtained is clllled Neumann's Bessel function of the second kind of zero order. Obviollsly if we ndd to Yo(.t) n function which is a COIl-"tunl multiple of Jo(z) the resulting function is nlso a solution of the differential C
(30.13)

106 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 30

In particular the function Yol')

" (Yol') - (log 2 - y)Jol.)) =.::.

n where i' denotes Euler's oonstant, will be a seoond solution of the equation. Substituting from equation (30.11) for Yo(z) in this equation we obtain the expression (_1.-&)' (I)' ~(,) (30."') :t l't r-I r where !p(r) is defined by e'luation (30.12). The function l'o(x). so defined is known n.s Weber's Bessel function of the second kind of zerO order. Thus the complete solution of the equation (30.13) is <)

Y,l') ~.::. (logl;')

I)

'"

+ y}J,I') _':':E

y = AJo(x)

+ Bl'o(x)

(30.15)

where A. n arc arbitrary constants and Jo(x), l'o(x) arc given by equations (30.8) and (30.14) respecth!c1y. It can be shown by an exactly similar process that when v is nn intcgcr the complete solution of the c
+

where A, JJ are nrbitrnry constnnts, J. (.x) is defined by equation (30.S) lind Y. (x) is givcn by 1 _I (11-r-l)1 (") ....2' I .::. r a:

I)

y.(.,)~':'(y+log(;.))J.(.)--:E 71:.-0

l't

1 '" (-1 t(b)r+2.

--:E

:r.-o

'(+)1 r.l' T

(~(,+,,)+~(,)).

(30.17)

The function l'.(z) so definC(JI) reduces to the Yo(x) of C(luatioll (30.1.1) as v-+ 0 and is known n.s Weber's Bessel function of the second kind of order v. The varintion of l'o(x) and Yl(Z) for n range of "alucs of x is shown grnphieally in Fig. 10. I) This function is denoted lU "''.{%) by Coumnt nnd Hilbert.

BESSEL FUNCTIONS

/

101

j',lr)

o 1-+--01-~--+-.w,c-7.G--7"'--,~P--:'IO' -x~

Fig. 10

VRrintioll

or

\".(.1') nnd

l~l(;r)

with

:.1:.

The functions J. ex) and l'.(z) nrc independent solutions of the equation (30.1). but in certain circumstances it is ndvunlngeous to define. in tenus of them, two new independent solutions. If we write

+ iY.(x)

(30.18)

J1~I)(.l:) = J.(x) - il'.(x)

(30,19)

lJ~I)(%) = J.(x)

then it is obviolls thnt we ctln lake the gCllcrnl solution of Bessel's differential cquntioll (30.I) to be y = A11I!l}(x)

+ A~Il!~)(x),

(30.20)

where Al nlld A, nrc arbitrarr constants. The functions U!ll(z). Jl~21(x) defined by C
108 THE SPECIAL FUNCTiONS OF PHYSICS AND CHEMISTRY § 31

the same diffel'entinl cqufllions and recurrence relalions as the function .I. (.:r). 31. Spherical Bessel Functions. A problem which nrises in mnthemnticnl physics is that of t.he solution of the wavc equation in spherical polar cooniinatc.<;

a'lV'2+ 2 aVI + _1_ ~ (Sill 0 alp) + ar r ar r 2 sin 0 ao ao

2rp = 2.- a2rp. a r 2 sin 2 0 arp2 c2 at'!. 1

(31.1)

If we take

II

solutiun of this equation of the form lp = Y m.n{O, rp)v,(r)c1a'l.

(31.2)

where Y ,(0, p) is the SllTfllCC spherical hnrmonic defined by cquntto'n (23.3) and 'I,(r) is 1\ function of r nlonc which salisfics the equation J21j1 2 d'l' 11(11 1) I w2 I (Ir 2 + -; (f;: r'J. lJI ~ lJI = o. (31.3)

+

+

Now putting

lJI=r-ln we see that equntion (31.1J) become;; (II +.. ~j21 R = 0 c2 r~ whose general solution is readily sccn to be

iF1! dr~

+2- dR. + {W'J. _ r (Ir

II = AJ n+1(wr/c)

+ iJ.I_n_i{wr/c).

(31.5)

Hcncc the function '1'= l'm,,,(O,PP±f"+il(wr/c)cl " l (11I.6) is a solution of the equation (31.1). The fUlIctions .I ±In+ll{k) whil'h occur in thc solution (31.6) Ilrc called spherical Bessel functions. We shall now show that they are relntcd simply to the circular functions. l~irst of all we consider the Bessel function (re). If we let II = ! in equation (30.6) lind mnke Ilse o the duplication formula. for the gHllllna function wc obtain the result

.II

131

BESSEL FUNCTIONS

'09

(81.7)

=- !

Agnill. if we pill" the relation

in equation (30.6) we obtain (31.8 )

The Q1hel' fUl1<:liolls J",(x) where 1/1 is hnlf all odd integer Illay be worked Ollt in n similar fnshioll. It is left llS lin excrcise to the render to show that

J .. (3') =

l:)'U..{x)

sill x-

g..(x)

• I

J_",(:r) =

L:.;:r) (-

I

)"-l{g",(x) sin

where the functions /",.

oos x}.

:£+/ ... (:r) (.'
t:", arc gh'cll ill Table I. Tnblc

'"

I.•

5

3 .T~- I

,

15

7

3

" II

.,

u

15 ",,-

105 ~1"

105

-15 --+ ~

10

r--~

015 _105 + 1 -",- - -,.- + -:r .r' ,i! ---------' !H5

42u

J5

110 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 32

These functions which arisc in thc way describcd have bcen tabu[lltcd in 1'nhles 01 Spherical Bessel FUlictions 2 vols. (Columbia Univ. Prcss, 1!)47) prcparcd by the Mathematical Tables Project of the National Bureau of Stnndards. 32. Intc~I'nls involvlll~ Besscl FunClions. Tn this section wc shnll derivc the valucs of some integrals invoh'ing Bessel funet.ions which nrise in practical applications. Tn the first instance we shnll eonsidcr definite integrals. From equation (27.5) we hnve the relation

J:'l:"J ,,_1 (x) dx =

[x"J ,,(x)): .

If tl > 0, x".!,,(x) --)- 0 ns x --)- 0 so that the [ower limit is zero and we obtnin the integrlll

,

fOiV".!,,-l(Xjd,'1:

= «".1,,(<<),

('/I> OJ,

(:32.1)

which, by n simple changc of vnriablc, gives the result f:r"J "_l(~r)dr =

~" J,,(~n,),

(n> 0).

(:32.2)

A partieulnr case of this result which is of frequcnt IISC in mnthemalicul physics is obtaincd by putting 11 = 1 in equation (:32.2). In this wny we obtain thc intcgrn[

" forJo(Er)dr = T J1{nn (l.

(32.3)

Furthcr results may bc obtaincd from (:12.2) by familiar devices sllch us integrnlioll by parl~~. For exmnple, making use of equation (27.5) we may write

0,21' or {rJj(Er)} dr, f't.!o(~r)(/r = fOla anti, integrating by parls, we sec that lhe right hnnd side of this equation becomes

132

BESSEL FUNCTIONS

a'

'"

... r"

eJtJ·Ur)dr

~Jl(ea) -

which reduces, by virtue of 03

(32.~),

Lo

2a!

TJ,(,a) -I'J'('a). Now by the recurrence relation (26.2) we hnve the c.,\,:. pression

" J.(';a) - Jo(;a) J,{';a) = io so that finally we have the result

f:r3,J o(Er )dr = 2;'! { J 0(';0)

+ (~(le - (~.;) J I(0;) }. (32.,~)

Combining this result with equation (82.lJ) we obtain the integral

f

.

40

/(0' - r').Io(Er)dr = ~JI(';a) -

'")0 2

-,p Jo(';a).

(32.5)

The most commonly occurring infinite integrals nrc 1Il0st easily evalullted by menns of substituting the formula (27.3) in parts (ii) and (iii) of cXlIIllplc 17 of Chapter II. From pnrt (ii) of that example we sec that a· 2'r(1,+1)

fa:>00 F I(I' + I',

refl+V+

_jn2x2)X"+pc-P:t.(/x

I )n' ( 2. r(v+ I )p"'+rit !F1!!I+!V+!,

=

!.,t+!I+ I; v+ 1;

-

a\

p:)"

(32.6) If we make use of equation (:l0.8) on the left-hand side of this CfJlllltiOIl and of equation (7.'Q 011 tile right-hund side.

we see that this result is equivalent to the formula

F(p+l+ t )(1"

a:>

fo J • (ax)XI'c-P"'I!X= 2" F(I'+ 1 p2)il'+Ir+i ~ .' ) , f', (!/l+b+l; h-l/lj .. + ._-- v+lj a-p)((J~+

where p

>

0,

,II

+ > o. V

X

(32.7)

112 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 32

The hypcrgcolllclric series occurring 011 the righl-hnnd of this equation nssumes tl pnrticularly simple form if either I' = )' or I' = .' 1, find we obillin the formulae

+

'"" Z' r(v+!) a· So J.(az):.r!"e-uth:= rei) - . (ll~+p~)*r

So oD

+ *l

va'

r(') - ' . . .. I ' (32.9) ! «(I.+~)~ of the forlllulu (:J2.8) which occur

J.(ax)X-+1r.-PZrfz=

Two special CIHICS frequently lIrc

2o+lr(••

(32.8)

" So Jo(a:t)r'z dx =

,\/(a'

1

+ p')'

(112.10)

a " • ."". So ;r.!.(oz)rJ>Zt[.z:= (a-+P-r

(32.11 )

IntcgTtlling both sides of C
"" 1 So .l1(UZ)cIllZdz=-a

IJ .f(' ')' a p v A special cnse of (:J2.!) which is oftell needed is Q

+

So oD

:z:Jo(ax)cl'zdx = (fl ' :1)-3. '") f:"

(32.12)

(32.13)

If we leLl1 tend to zero 011 both sides of c<)u:llion (32.7) lI'e filld nUll we Cfln sum thc hypcrseometrie series by Gnllss's theorem (1.2) pro\'ided lhaL JI < 1 We

I I I,' + I.

lhen havc the rcsult

I 2"r(i + !/l + !v) fl"+ll'(! !/l+ !v)" S", ( ,- (x=

(3'_'.1.1)

0 ' · (IX1""

Similarly rrom part (iii) we haye the equation

f:

oF1{1'+ I;

or

e:mmple 17

or

Chapler lJ

-i n!,r)cp2z1 »-+-I dx =

+

r(-!p ·bl . ( -21'''-· - ,'., ./,+h; -.

1+ I;

(/!)

-41),

133

"3

BESSEL fUNCTIONS

which, because of

(~7.3),

is CfJui\'illcnl to

I~J.(az)cplz1x"-ldz

+

a·r(!f' !JI) ,( "') = 2*1p"""'r(I,.f-l) 1/'1 !P+!J'; v+lj - 4p!' (32.15) From parts (i) and (ii) of Ex. 11 of Chapler the spccint cnses

J[

we hn\'c

("'.10) and

Jo"

u' ) r"l/b"(32.11) 1r!'+sJ.(a;r.)e-P'="r/x= .... 1u·~>+I (1'+1--, 2 1)· ,~p

of which the

1lI0st

frequently used

l\rC

(:1:!.18 )

and (32.19)

33. The Modified Bessel Functions. By nn argument similnr to tJwl employed in § I we enn Nlndily show that Laplace's equation in cylindrical coordinates

;r-~ .L - ' iJ~ + -' ;r-~ + ir"/' _ 0

at!'

!!

at!

possesses solutions of the

,i apt

a::.'--

(Oflll

tp = eH..±f··R(e) where R{!?) satisfies the on:linnry differential equation

CPU + __

de':

1 dR

e de

Writing x in place of

111(1

( /112+_ ,,) U=o. (J2

("".1 )

we see Ihat this cquillion is

114 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 33

equivulent to the equation d~R

11x2

1 rlR

1,2)

(

+x- (/;z

1

-

+ XZ u= O.

(33.')

If we proceed in cxnctly the SlIllle way us in § 30 we call show that if). is neither zero nor nn integer the solution

of this equation is R

~

AI, (x)

+ BL,(.,)

(33.3)

where A and n nrc arbitrary COllstnnts and the function J. (x) is defined by the cquntion tl!

'"

U..&y

~

x·

..

I.(.x) = 2rr(v+ 1) r~o r1(v+ 1 l. = z-r(Ir+I) 0/' l(V+ 1; ix-). (33 .•') Compnring cqllntion (tWA) with equntion (30.8) we sec thnt 1.(x) = ';-rJ.(ix)

(33.5)

result which might have been conjectured from the differential equution itself. If l' is on integer, n say, then 1_,,(x) is n lTlultiple of I,,(:I:} so that the solution (lla.3) in effect contnins only aile arbitrary COllstnnt:. Dy II procc.<;s, similal' to that outlincc! in § ao we eUlI show that: in thesc circumstanccs thc gcncral solution of cqull.tion (a·k2) is

1\

R = A1,,(x)

whcrc the fllnct:ion [("(x)

f( ,,(x)

+ 11K ,,(r)

(33.U)

is defincd by the cqllntion

= (-1 )"H 1.(x){log (tx)+ y) + '* "il (-1 )'(11- r-1)1 (Ax)-"W -r-G

r!

-

+ 1(-1)" r_lr.11 ~ 1(. ~ r )1 {p(,)+p("+')H!x)'+" The functions J ,,(.:1"), K,,(x) defincd by cquations

(33.7)

(a}:.~)

133

BESSEL fUNCTIONS

115

o

nnd (&f.7) respectively nrc known ns modified Bessel (unclions of the first and second kinds. The rcsult (:.l3.S) is very useful for deducing propcrlk-s of the modified Bessel function 1,,(.%) from those of the Bessel fUllction J ,,(x). For instance, when Ii is nn integer it follows from C
(:13.8 )

find from equations (2G.1) to (26.5) rcspccti\"cly thnt 21~(.1:) = In_d.'!:)

+ '''+1(x)

2/1

- I ,,(x) = J n_d:r) -

(:l:l.n) (:13.10)

1II+I(x)

• xJ~(x) =;r[ n_t(x) - 1/1,,(x) xJ~(r) = III ,,(x) + .r1"+I(x)

(33.11 ) (33.12)

10(x) = J1(x)

(33.13)

l·U.--~--~-~--~--,----,

0··1

o Fig. II

2

:1

-x_

Variation or r-/.(.:r).

t-'/,(~)

-I nnd C-/1(:l:) with. z

116 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 33

Similarly equotions (27.5) and (27.7) imply the relations (:l3. H) (33.15)

All of these relotions call, of course, bc dcrived dit'celly from the definition (33.'1) of I. (3') nnd it is suggested ns Ull excrcisc to thc readcl' to r!ct'i,"c them in this wily. It should Illso be observed thllt K,,(x) sotisfics the Sllille recurrence relutions as 1 n(x), 5 ,,-r---,rr---r--~--~---~----,

3

o Fig. 12

, Vnrinlioll of

5

3

-x_ ~f(.(;r), e-K,(.1:)

nnd

~K,(ar)

witll ;r.

The vnrintion of lu(x), 1 1 (a:) nnd l~(a:) wilh a: is hown graphicnlly in Fig. Il nnd that of [(0('1:), K1(x) nnd A·~(:r.) is showll in Fig, 12.

134

117

RESSEL fUNCTIONS

34. The Bcr and Dei FUllellolls. saint iOllS of the form

If we wish to find

IJ1 = U(e)"i"'l

of the diffLl"ion equation a~V! U(!2

I

u'P

I

uy!

+ e ae = -;; at

we hln'c to solve the oNlinnry diffcrcntilll equlItion /PU I till iw /' +--1=0 df!2 f! de %

.

On chnnging the independent vfll'jnble to:1: = (w/>::)i(l we sec that this Intter equntion is cqui\'nlcnt to the equation

ir-n + 2- lill _ dz2

ill = O.

x dx

(3'1, J )

Formnlly we mny lake the independent solutions of this cqul\tion to be loUix) lind A·o(iix). I{ch'in introduced two new functions bcr(x) and bci(x) which III'C respe!:tin:ly the real nnd imngillnry purl" of [aUb:), i.e. bcr(x)

+ i bcl(x) =

Jo(ii.r).

(:},~.2)

From the definition (33A) of 'o(:v) we sec thnt ~

(-1)'(l>')"

.-0

-$.

bcr(x) = 1,

(" ' ) 2 '

(:H,IJ)

. ~(-I)'(iX~)~'+1 ' hCI(ml=l. ("+1)12 .-0 _9 .

(Il.L.q

Hnd lhnt

The vorintion of t.hc functions ber(x) and bci(x) with x is shown dingrnmllllllicnlly in ]~ig. la. In B similnr war the functions ker(x) lind kci(x) nrc defined to be respectively the real nnd imaginary pnrts of the complex function J(o(ilx), i.e. kcr(m)

+ i kci{x) =

J{o(ih).

(M.5)

118 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMIStRY § 34 3r--~-~-~-~--r---,

bei(x)

-I

-3

-5 -6

ber(x)

----"_----IJ

L-_~_~

Fill. 13

Vnrintioll of her{ot) lIIld hei(.:!:) wilh z.

From the definition (33.7) of Ko(x) we can rcndily show thnt kcr (x) = - {log (!x)

+ y} ber (x) + l7t bci (.'1:) + 1 )'(~,'I:)4' (<)

*(_

'::1

(2r)12

rp _r ,

(3·'.6)

and that kci (x) = - {log (-!x)

+ y} bci (x) ~

+.:'0

17t bcr (x)

(_1)'(!x)4r+2 <) (2r+ 1 )12 Ip(_r+l).

(34.7)

Fig. H sho\\'s thc variation of the functions ker(x) and kci(x) over a range of values of the indepcndent variable x. The four functions bcr, bci, kcr find kci lire IIse(1 marc often in clcctricnl cnginccring lhnn they nrc ill physics or chemistry. For a ful! account of t.heir properties lind thosc of their genernlizalion to higher order and of their appli. cation to enginec.ring problems thc readcr is referred to

ps

119

BESSEL FUNCTIONS

N. W. McLachlan's Bessel Flll/cliOllS (Oxford University Press, l03,q.

for

Engineers,

" c" kcr(x)

"

(J

/

c" kci(x)

'----~-"'~----:--/''!_-___:_-____;! :!:3'" 5 G

I

-x_

-I

Fig.

l'~

Vnrintion of

k~r(:l:)

lLnd kci(a:) with

;t.

35. Expansions in Series of Bessel Functions. know from § :;0 thnt

{

.,rdx~ +

IV'"

,j

oX

dx

,. + (t.-x

2

-

.,r- + Ir-/d + (/1.,1:. •• { x--/ Q

(X-

lOX

j J m(}.x) = 'j .1,,(IIX) = 1/-)

m:!)

We

0,

(35.1)

0,

(35.2)

so that multiplying equation (35.1) by.1,,(/lx)/x, (35.2) by Jm()..r)/x, integrating with respect to x from 0 to a and subtracting we find that (i.2 - f/2) fa;rJm().x).I ,,(px)d..c

o

+ (II:! _

Ill:!)

f°.! m().x)J 0

,,(fiX /x X

= a[p.J ,,(J.a )J;"(IUJ) - J.J ,,(flO )J;"(}.a» , (35.3) if11>-1,1II>-1.

120 THE SPECIAL FUNCTiONS OF PHYSICS AND CHEMISTRY § 3S

Putting'

fo":r.l

11I=

n in this rcsult we find that if ). #- II,

lI().l:).IlIVI;r;)dx= ... a .. r,ll ,,(J.a).I~(,)(l )-U ,,(/la)J ~(}.{j)]. 1.--/1(35A)

J.

Thc corresponding exprcssion for = p is obtained by putting It = ), e, WhCI'C e is slllall, using Taylor's thcorem and then letting e tcnd to zero. We find that

+

I:a:p ,,(i.x>pdx=~a2[ {J ~()'(llF+( 1- J.~(:2)] {J ,,().a)}2. (ll5.S) Suppose now that J. nnd II arc positi\rc roots of the transeendcnLnl e(}tllltion hJ,,().a)

+ 1i).{jJ~(J.a) =

0

(U5.fj)

where h nnd k are constants. 1 t follows than that

J:.T.J"P..t:)J

lI

(/ltC)d.1J = CAOA.i"

(:J5.7)

where CA = {J ;~~;'~}2

{J.-~).2{j2 + 112 _

If wc !lOW suppose that we can expand functiou I(x) in the form I(~)

~

=

, a.,J

lI

(35.8)

k2112}. all

nrbitrary

()./x)

(35.!l)

where the sum is taken over the positive !"Oots of the equntion (35.G) thcn wc can determine the eocffieienL<; "I as follows: Multiply hoth sides of equation (:35.0) by .:r.1,,().,x) find integrate with respect to II.: from 0 to 11 thcn

f,"xj(x).l ,,(}.j1:)dx = L, a f".T..I , ,,{i.;x).1 "p.r j

'!.: )r/;r;

from which it follows that a, = - 1 CA,

f" x/(x).1 nV-F) dx 0

(35.10)

121

BESSEL FUNCTiONS

\36

Because of its similarity to a Fourier series a series of the type (a5.!») is called a Fourier-Bessel series. In particular if the sum is taken over tile rooL.. of the equation J;(!.a) ~ 0 (3:'>.11 )

then the coefficients of the sum (35.n) flrc given by fl J

2J·7

= {J (' n

"Jo

»)2'

2 ~1

(}.J

~

/1")

0- -

f";1:/(:r:)J,,(J.p:)clx 0

(35.12)

Similarly if t.he sum is Luken over the positive roots of the equation J,,(i.n) = 0 (85.13) we fiud lhat the coefficients

(I J = 2{1.t. n . "

{lJ

lIrc given by the [ol'ollila

Wfa:r/(X)Jn().sX)dX

I'lll

(35.H)

0

In this section lIO tlttcmpt has UCCIl made to discllss the VCIT difficult problem of the cOll\'crgcncc of FourierBessel series. For l\ vcry full discussion of this topic the render is referred to Chapter XVI I r of G. N. Watson's A 'l'reatise on the Theory 01 I1cssel Functions, 2nd. edit., (Cambridge University jJ ress, I!H.q. 36. Thc Use of Bessel Functions in Potcntial Theory. As nn example of the use of Bessel functions in potential theory we shall consider the problem of determining a fUlletion tp(/.>, z) for t.he hnlf-spuce fI :2 e :2 0, z:2 0 sutisfying the differential equation

a2lj! I alp aZlj! oe' + e oe + 0" ~ 0 and the boundnry conditions:

(q (ii)

•

,~lle),

on

,~o;

'1'-> 0 as

:J -i'"

co;

(30.1 )

122 THE SPECIAL FUNCTIONS Of PHYSICS AND CHEMISTRY

(,.,..,) a, ae + xrp = (iv)

0

i

36

on f! = ll;

e-+ O.

'P remains finite as

We saw in § 1 that a fUllction of the form 'P = R(e)Z(::) is n solutioll of IXjuntioll (36.1) provided that

cPZ dz 2

2~

).,Z = 0

-

(36.2)

find that (/21l

1 (Ill

( Q-

Q (!?

-1.+--1

.2]"' +1.j~=O

(00.3)

wherc).( is n constant of separation. To sntisfy the boundary condition (ii) we lUllst take solutions of equation (30.2) of the form

and to satisfy the condition (iv) we must lake as the solutions of equat.ion (36.3) functions of the form R = JO().&I) since the second solutions YO(),ie) would become infinite in the region of the nxis e = O. The differential equation (36.1) and the boundary conditions (ii) llnd (iv) urc satisfied by nny SlIlllS of the form 'f'(r, =)= :Ea1c-Aj'Jo()"!?)

,

(BOA)

wherc the a/ and )./ nrc constants. But if we arc to $illtisfy the boundary condition (iii) we must luke thc sum over tbc posili\"c roots of thc equulion I)

+

)'/Jo().,.a) xJo()".a) = O. (36.5) The solution is dctermined therefore if we elUl find constants a/ such that condition (i) is snlisfied, i.c. such thnt

,

!(fJ) = :EatJ(}.;!?)·

(30.6)

1) For propcrliC!l of lhe rools oUhi! c{llmlion!l« exnmple 16 below.

BESSEL FUNCTIONS

§J6

From equations (35.10) nnd (35.8) we sec thnl we must take t).2

il =

_J.,

+ ;.,:2)

ll'!().~

j

f n'J(n')J ()..n'}tle' (Jo(J.,a)}2 It

0<:

•

(30.7)

o,~

llcncc the requin.-·d solution is !p(r. =)=

2~ ~ .~.~CJjlt°(j~i'!)} .r:()'!(Q')Jo().,Q')de'

il-

f

( ....

i+x'!) Jo(J.,fI)

2

(3G.8)

0

where the slim is luken over the positin~ rools of the equAtion (3G.5). If, instead of the ooulldnry condition (iii), we hnd the condition 'I' = 0 on e = a then it is cnsilr seen thnt the solution would have been ~

.-,

...-l,IJ (' hi

_.:... ~..

~(".) -

(I

0 "I"': , ~I {J I (J.,-11 I)'

f' eI(e IJ,('·,e Ide "

'.

,

(30.9)

0

where the sum is tuken oyer the positive roots )'1 of the trnnsccndcnlnl equation (30.10)

For exnmple suppose lhnt 'P satisfies the conditions '1'= 0011 (! = il, 'P-O as .:-+ co, 'P= '1'0(0 2 - Q'J.) on :: = 0, 0 s e ~ a, then the solution to the problem is gi\'cn by c
r'".

•

'.

•

J"eJ(e )J.(J.,!! ),Je = o

.laro.

2flZ~~o.

.:1 J.(/.,a)- ~ J o(.I.,a) .I./.I.(

"tlV'O

•

= --:s--J 1(.I.,a)

'.,

since J.( is n root of equnlion (30.10). Thlls the rccluircd solution is (3(;.11 )

124 THE SPECI....L FUNCTIONS OF PHYSICS AND CHEMISTRY § 37

Tables of the first fort)' zeros ~( of the fUllction Jo(~) with the oorrcsllOllding values of JI(~') are nvnilnblc l ) so that it is COIwcnicnt to express results of the kind (M.II) in terms of them. It is rendily seen that in this case IJ

where

C= zla

.(

_) _ 8a~.' )~ rll'Jo(lX~,) 'fa 7 ~Jl(~tl

r,. -

and

IX

=

(36.1"_)

eta.

37. Asymptotic EXllunsiolls of Bessel Functions, In certain physicnl problems it is desirable to know the "nlue of n Bessel fUllction for large vnlucs of its argument. [n this section we shull dcri\'c the nsymptotic expansion of the Bessel fllllClioll of the first kind J ,,(.x) and merely imlicnlc the rcsulls for the olher I]csscl occurring in malhe· Illnticnl physics. We L'lkc equntion (~~.1) as Ollf definition of the function J ,,(.2:). Applying the theory of functions of n eomplc.-.: variable it is readily shown that this dcfinition is equivalcnt to J (.2:)= _ _ (1.2:)"

"

rmFln+1l

{f

c.

(l-t')"-i~I~'{U+f (1_r.)-I~IZjdt} c.

(37.1)

whcrc L 1 is llle straight linc 9l(t) = - t in the uppcr luJlf of eOlliplex: (-plnne and £2 is 1I1c corrcsponding p:trt of lhe strnightlinc !Jt(t) = I, By dwnging the \'Ul'jllblc from t to II = ix( I - I) in the first intcgr/ll and to II = - ix(l - t) in the seeond we ..ee that

+

(31.2)

I) A. Gnly, G. n. )laUlews lind T. lU. Mac.Robert. A Treali,fe 011 Bend Flmttioll,f and Their AIJplitlllioll,f 10 Phy"iu, 2nd. edit., (Macmillall, 1031).

137

'"

BESSEL FUNCTIONS

and i:(x) denotes il~ complex conjugntc. Expnnding (1 juI2:r:)"-i by the binomil11 theorem lind intcgrnting term br term we find tlml

+

;,,(x) = !c'''-lioo+ll''l ~Po

(1 + II,

!-

II;

'.!~x).

(37.3)

Tf we ndopt lInnkcl's com-eolian of writing

(II,

r) = (_I)" H -

II);:! + Il),

in equlltion (:17.3) lind substitute the result in cqufl.tion (37.:!) we find finully Ihnl for lorge nllues of x the asymptotic expansion of the Bessel (unction J.(x) is J,,(x) .......

-

fT

{ • (-1)'(" ",) r-=<.'OS(X-!II:r-J:r):E (:!,z:t .r ' - nor ....0

l

,. (_'!":r- 1\ ~ (-1)'('"I!'+!,,+ 'I} . III

:t, ,-
"" -

("_oX

(ll7..')

The corresponding expansion for the Bessel function of the sc<.'Olid kind is fOllnd to he Y,,(x)......, 10(Sill(X_'l\Il;1l_ in)

V;x

-

~

(-

r-O

~t(;~, 2r) + 2x·'

"'; (-I)'(1I,2r+I)1 +cos(x-ll1n-i71) 1. ()"+l . (37.5) -

,-0

:!x

.r

Substillltirlg these nsrmptolic expressions in equations (SO.IS) nnd (30.19) we find that as.1:_ co,

Il~ll(x) ...... r..!!....)i i~(x), ~7X

1J~2)(x) ...... r.!.)ii~{X) \~X

(37.6)

where i~(.x) is given hy equation (:n.a). 1n cerlaill problems ollir a "cry crude approximation to the beha"iour of the Bessel function is desirc<1. III these eircumstllllccs the following forllllllae are usually suf·

126 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

t

37

fieient:J ..

(Z).-.l~COS(Z-lml'-t:T).

Y..

(Z}.-.l~Sin(Z-lmf-i:t); (37.7)

1l~II(z).-.l/2 e''I'-i.....'-J"" 11~t)(Z).-.V 2 e-l~l ..... Hi"'. Y7lX :rx

(37.8)

Similar formulae exist for the modified Bessel functions. Proceeding ill tlle samc way liS ill the establishment of equlltioll (28.1) wc eUIl show that 1 In(z) = .•/:rr(1I

+ I) (Ix)n J'-t:'I"(1 -

t2 )";,1t

which becomes I (x)= n

I

,/(2rrz)r('I+H

{e--IIl+ll"J"c"l,n-l 0

+ e'" f~

(l+~)n-ldll 2%

rIO Il

n-l

(1- ~~) n-J

(ill }

by a simple ehangc of \"tlrinblc. Oy n method similnr to t.hat cmployed nho'·c to obtnin thc usymptotic expunsion of ./ ,,(x) wc eun thcn show l,hnt if - b: < nrs z < 1

'" ~ (-l)'(ll,r)

l"(.)~ \ /(._:TX )' ,-0 l:

-i-;z;

(')' _.1:

-

*7t, -

c-",+(n+!lnf '!:. (tl. r)

+ -v_'(' ) ,_0 l: (.-x )' ~rrx

nnd that if < arg z
+

"( )~(~)I~'I' ~ (II, r) x ..._x ~ r-o_X _(")'

n. ..

as x -+ 00.

BESSEL FUNCTIONS

127

EXA!>II'LES

MlIking use of Example 2 of Chapter II and of the expallsioll expaud cos (:I: /jin 0) RS n power serie.'! ill .'lin 0 in two WHy.'!. lienee hy equaling powef:5 of lIin"' 0 IIhow Ihat if Il is a positive integer

1.

(~5.8)

:1:" _ :!"+1 ; (n + Il - 1 P Ju(x). n_. (n Il)!

Derive the corresponding result for x"·t! rmd show thnt the two resull!l IlIny be comhined into lhe single formula ... ;~: ( r + I l - I ) ! ( r x' __'

n-o

+ O)J _11 .,u () or

"I

(r_l,:!,ll, ... )

2. It" lind show Umt

C dellOte

.

t.he eccentric and mean nnOllllllie.'l of n plnnet

cos ('IlI) _

/I

sin ('Ill) _"

,

1:: _.I.. _.(me) cos (me) m__ t
!: -.:..m J .. _.(me) sin (mC) .

...__a>

3. :'>lllking \l$C of the eXllression for p. (cosO) given in EXIIIIlplc 19 of Chnpter III show th"t

. !..... ~

n..()

4.

Show lhat 8.1;'(:) _ .I._ I {:)

(i)

.U~"(~)

(ii) 5.

P .(cos 0) _ trc:os8J.(r sin 0).

"I

-

3J._,(:)

+ a./~(~) + .1.(::)

+ 3J.,,(:)

- J.,,(::).

_ O.

Prove thllt

_+ ' .(r.1: ) , ~_ { - l)J :!

_ S"'COS(l\,+i)lI -=~"'~'--c;" (_1)"" 1I 0 cos lu ,{(X'II')

r-l

nnd dcdu(:e tllat

_• + •~ :!

r-l

(-l)'.I.{rx) _ O.

128 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY 6.

Shol'l" lh.Dl lhe 21 _

with freedom equatiollll t - sin f, y _ costl(l - cos I)

C'l1I'.... e

may he rCl'fell(:I11ecl ill lile inlerval 0

<

I

<:¥ by the Fourier 5Cries

•

y _ 21: J.(u) ros (liZ).

0-'

7.

I'ro\'e that tilr ...... _

(~)l ;: (~r1+ I )ell'"'' J.+l(I:r)P.. (eos 01. 2kr

'J.

..-0

Show thnl

i

(ii)

,,-0

(-z)· 0-1-4 .. ./...+,,(~ va) _ 'll

(21+11)-1 ... J .. {~ v(z + 01).

". II.

12.

If

!It

>

0

show Uml

f: eo ... •

cos(z sill 0)diJ - J.( ,/(21' - a'))

P"we lhat, it - 1 I

<

z

",;;-,,,,,.. _ l:¥ v(l

Deduce thnl

.:1:')

<

1.

+::

•

E .1,(111::) COlI (.n."Izl

_1

'"

BESSEL FUNCTIONS Pro\"l~

13.

Ulal ,

...

'

J.{z)J_.(z) - J.(z)J_.(z) EO-

•

where A b n consl:lIll, 811d, b)' considering the series for J.(z) oml J.(z) when :r is small show that A _ (2/;t) sin (r:r). 14. Show that thc colllilidc solulion of Bessel's Cflualion lIlay be writtcn in tllc form

where II lind JJ nrc nrhilrnry const..'HII$.

15.

Slio\\' lhal the complete solution of lhe differential C'IUlltiOIl IJ'y I _+_zy_O

""

i' where

,p _

'~/:!7

,

and II IIml lJ nrc orhitmr)'

eon~l.'ln~.

16. If (J nnd b nrc felll cou!tnnu show 1lL,.llhe tOOl.ll of U1C equntion (lXJ'.(z)

+ b.l.(z)

_ 0

life simple fools e~rCllt possihly lhe root z _ O. Show "Iso Uml the Cllllulions .1..{z) _ 0, .1'.. (r) _ G hll\'e no roo.., In (:01llmon ex<.-cpt I'o.• ~ihly II: _ O.

17.

If z > I lllld

til

+" +

1 > 0, prove thnt

f:C-·'J .. i(I)'.--1flt-l/~ where Q=(z) kind.

denol~

the fl.SIOCillted

c

l.(.c +y) -

~ l ..{.cl/ .._..{y) ..-41

(z'_q-i·Q;(z)

I~gemlre

tunetioll ot the 5eCOnd

_

+

~ (J ..(z)/ •••{y) ... _1

+ l .....(.cll.. (yl)

130 THE SPECIAL FUNCTIONS OF PHYSCIS AND CHEMISTRY 19.

Show that

10.

Prove that

..

-(t-

~

" _ _ Ill

./.{la)l· _ e-I'

I )

..

..

~

.. __ 00

1.··t·J.{~)

lind deduce thnt J.{re'8) _

(i)

..

~

J •••(r)e/'·+"l8

(-ir sin 0)· ,

.. -0 l.(~)

(ii)

_

....

e.-

:E - , Ju.(z).

"'-0'" Using Lhe expllnsioll of the la~L (llll:stion prove 1I1l1L J,(ae''''+be'j!) Is the coefficient of t' ill the expnn~ioll of

21.

eXj' { _

+

i(a !lin a. t b lin

(I)} __.. i: _ ;: .. e'.':H'"",J.(a}J..(b)I.'.

+

Df putting If _ a cos a. boos{l. 0 _ alina. prove XeUJlIIIIIl\', additiOil theorem J.(ll) where Il' _ Il' 22.

+ b'

'

~

+ bain{J.{J-a.+(J-:;

_ /.Ie-'& in ...

be

'8)

__'" E .I•••(Il)J.(b)e- lod

- :lab cos O.

Prove that ".(=}J.(Iu:)

where /I' _

(I'

+ /)' -

_...!.. I J.(U.r)dO, :J Jo

:lu/) cos 0 ami deduce thllt

where 1:

Pro"C, ill " similar way, thnl

1

-

4a'

(Il

+ 1t)1 + c'

131

BESSEL FUNCTIONS where I: find N(kl are

f

l"

0 ,/(1 - !;l s ill''1')tlrp

/:;(1:) -

13.

defined lIhove and

liS

Show that

J ..(z)J ~(z) _ .:.. ell~-.l!T/ fn. 1~ __ (:!.e sin O)eIR+.. IOI dO

,

"

nnd hel1t'e tlmt (JR(Z))" _ -

Deduce lhnt

24.

, f",

J u (2xsinO)dO

"

Prove lhal

25. We ddine lhe Bessel-lnteJ.1rlll runctlon or order II by lhe e(lliation

"".1 R(Il) --1111

f,

./iR(:r) _ Prove t hul ir " is evell

_.2.-

"

rem

(1I0)ci(zsin 0) 110, :r 0 where ci(.r) denotes the cosille illlegr:!!, Pill! derive the correSl>onding expression wilen " is odd. Show that:,/iR(Z)

(i) (ii) (iii)

(iv)

(v) (vi)

Ii (e!·"-l/") .. Ji~(z)

•

~

J

I"Ji.(x):

_ J .(z)/or;

+

(II - 1).I;._,(z) ~ (II I )./i~.,(Z) _ 2ItJj~(z): ci(:e) _ Ji.(ie) - ~./i,(z) + 2./i.(z) :

ai(z) _ 2./i.(z) - 2./i.(x) Ji.(z) _

r + logHz) -

+ ~./i.(z)

-

;

"8,1".(1, 1,2,2,2, -z"/4).

CIIA1''rEll V

THE FUNCTIONS OF HERMITE AND LAGUERRE 38. Thc HCI"mltc Polynoml:lls. The Hcrmite polynomial lJ"(x) is defincd for intcgrnl valucs of Il and all rcal vnlllcs of oX by the idcntity _,21~_1'

I,

__

~ lI,,(IV) t"

..... _ _ ,_.

,,-0

]f

(38.1)

"/I.

we write

thcn it follows from Taylor's theorcm thnt.

1I ,,(x) ~ (O"~) at ~ ,., [0: at ,-I-"'J (.. 0 1-0

Now it is obvious from the form of thc function c:.:p {- (It - t)2} that [

0"

_

at"

J

C-IZ-I)"

= (_ 1)" _d" (c-",)

dx"

1-0

nnd so wc havc thc form

d"

//"(x) = (- 1)"C"I--' (r z ') IX" 1

(as.2)

for the cnlculntion of the polynominl //,,(."1:). It follows from this formula thnt the first eight I rcrmile polynominls nre://o(x) //l(X) //2(X) //3(X)

1, = 2;1:, = ·k/:2 - 2, = 8.v3 - 12x.

=

§38

133

,THE FUNCTI ONS OF HERMITE AND LAGUERRE

lf 4 (x} = 1I~(x) = 1/ 8 (x) = 111(x) =

+ +

12, lOr' - ·ISr 120x, :J2z$ - IOOr 04x' - .ISOr; - i20x' - 120, :J:JOOr - lGSOx. 128z - 13Hz" '

+

In general we have

II ,,(x)=(2 .:r)" -

"(11-1 ) (2X)"-2 11

+ 11(1l -

1)(1l - 3)(11 - ,~) (" )04

_x

2!

+ ...

or, ill the notatio n of Section 12,

1I.(x) = ('lx)"

~Po (- !", ! - 1"; - ~).

(38.3)

Recurr ence formul ae for the Hermit e polyno mials follow dirccLly from the definin g I'cllllion (a8.1). If we differen tiate both sides of that equatio n with rcspee llo.t: we obtain the relation

• /I' ( ) 2tc~ZI-I' = I: ~ t n ..-0

II!

(rom which it follows directly thal 2/1/1 n_dx) = Jl~(x).

(88"1)

On lhe other hund if we differe ntiate hoth sides of the identit y (38.1) with respect to t we oblnin the relation 2{x -

I)c~'_11

IJ (x) ~ 1"-1 .. = 1: 1)1 ,,_I (11

which cnn be wrillen in the form 21: ~ 11,,(:1:) t" _ 2 ~ 11,,(:1:) 1"+1 = .. -0

II!

.. -0

III

to yield the identit y 2xll,,(.%) = 21111 11 _ 1(x)

1': 11,,(:1:) t'l-I t)l .._1(11

+ 11 +1(z) I1

by the idenlifi ciltion of ".'ocfficicnl.!i of I".

(38.5)

134 THE SPECIAL fUNCTIONS Of PHYSICS AND CHEMISTRY

i

39

Eliminating 2/111.._t(x) from equntions (8S.·Q and (38.5) we obt.nin the rclnlioll 1I~(x) = 'lor-lI,,(x) -

1I"+I(x).

(:.18.0)

Differentialing both sides of this identity we find that Jl::(x) =

2xll~(x)

and, by equation (:lS.4). 11::(z) -

+ 211..(x) -

1I~+I{x)

lJ~+I(Z) = 2(11

2xJl~(x)

In other words y = JI ,,(x) is ferentinl C
+ 1)11 ,,(z) so that

+ 21l1l ,,(x) = 1\

o.

(:.IS.7)

solution of the linear dif-

+ 21ly =

(38.8)

O.

39. Hermite's Differential Equ3tlon. We saw in the II\St section that II ,,(z) is n solution of the differentinl equl\tion (88.8). Heplncing the integer /I in that equation by the parameter I' we obtain Hermite's diffcrentild equation tFy

dy

dz' - 2x d:r

+ '.!I'y =

If we assume n solution of Ulis equation

•

y = 1:

....,

(".! )

o. In

the form

(/r:z:r+l1

and substitute in the equntion (3G.l) we obtnin the recurrCllce relution 2(r e- J') (••.2)

+

on cqualing to zero the cocfricient of x"'k'. Equating to zero the coefficient of .zq-:l we obtllin the indicinl equlltion

....

( ) e(Q - 1) = o. Corresponding to t.he root f! = 0 we hllve the recurrence relation

THE FUNCTIONS OF HERMITE AND LAGUERRE

2(r - v)

135

(3.... )

which gives the solutio n ( _ 21' 2 _ YI(X)- l,. 1 2!Z

+

2:!1'(1'- 2) _ 231'(11_2 )(v_'I) II Z 01 It"'.U

)

+...

(89.5)

is n consta nt. where Similar ly, eorrcsp onding to the root (! = 1 of the indicial equatio n we have the recurrc nce relation 01

2(r+ l-I') flr+2 = (r

+ 3)(r+

(30.6)

2) or

from whieh is derivcd the solutio n y~ (or )=0it ( 1-

"(,-I ) "+"'( '-1)(' -3)". 3!

z-

5!

+ ...)

(30.7)

where a2 is a consta nt. Thc general solutio n of ] Jcrmite'!> differe ntial equatio n is thcrefo re (30.8)

For genern l values of thc pnrame ler II the two series for YI(Z) and Y~(x) arc infinitc . Yrom equlllio ns (3!l.·1) and (39.6) it follows that for both series ar+z""

rr

we write c.xp (x2)

then

=

bo

•

"7"(/"

as r

~

00.

(30.0)

+ b~ + ... + b,xr + br+2zrH + ... •

br+!! "" ; b"

as r ~ CO

(30.10)

Suppos e thnt a,V IbN is equal to a consta nt y, which may be (3D.O) and ~mlfdl or lnrgc. then it follows form equatio ns

136 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY §"'O

(3!1.1 0) thnl. for large cnough vnlues of N. Q,v+'! ...lb.v+t............ y. In other words the higher temlS of the series for YI(X), y=(z) differ from those of exp (z2) only by multiplicnti\·c constnuts YI' Y:t. so thnt for large "alues of 1z I, y.(:c) ........ YleZ"' y~(.J:) ........ YseZ·

sincc for such values the lower terllls nrc llllimportnnl. We shull sce laler (Section 41 below), that in qUlIlltUnl meehllnics we require solutions of Hermite's differential equation which do not become infinite morc rapidly t.Il1ln exp(!x 2) liS I x 1- 00. IL follows from the above considerations that sueh solutions llrc possible only if either y.<:c) or Y2(X) reduce to simple polynomials and it is obvious frolll equations (3!1.5) and (~9.7) that this occurs only if I' is a positive integcr. For cXllmplc if I' is an even integer 11 we gct the solution

y{x)

=

(~!I.lI

cll.(x),

where c is n constnnt, by laking

)

= 0,

Q=

,,'

a l = (- 1 )1" ('iIl)1 c.

Similarly, if J' is nn odd integer (39.11) by tnking a l = 0 nnd

",~(-l)l"-l(;"

11

we gct the solution

2Jll

,)1'.

Hermite's differential equntioll therefore possesses solll tions whieh do IIOt become infinite more rapidly thun exp(ix)~ as I x 1_ 00 if lind only if I' is u positive intcgcr II. Whcn this is so thc rcquired solution of Hermitc's equation is givcn by equation (3!1.Il). 40. Hermite Functions. A differential equation closely related to Hcrmitc's equation is

~~ + (!.-x')V~ 0

(40.1 )

'<0

THE FUNCTIONS OF HERMITE AND LAGUERRE

131

If we transform the dependent "nrinblc from 'P to Y where 'P = ci""y

(40.2)

+

Ilnel put;. = 1 .!,' then it is readily shown that y satisfies Ilcrmitc's cqunl . Ul (an, I ). The gencral solution of equation (.10.1) is thcrdorc given by equations (.'0.2) and (39,S) wilh Y,(lV), !J~(.r) given by cqulltioliS (:J!).5) lmd (30.7) rcspccLi'"cly. The 11rgullIcul at the end of the last section shows lhat the equation (.10.1) posst"Sscs solutions which tend to zero as I x I ~ 00. if nud only if the pammclcr}. is of the form 1 211 where" is n po~ili\'c integcr. Whell ;. is of this

+

fonn the required solution of (.la.I) is II. conslfintllluilipic of the function IJI.. (X) defined by the equation !P,,(x) = cj"'/lll(x)

(40.3)

where Jl ,,(x) is the TIcrmilc polYliominl of degree II. The fUlletion tf',,(x) is clllled n Hel'mlte function of m-der n. The reeUITC'ICC I'elations for 1/',,(x) follow immediately from those fur 1I,,(;t). For instance equfltion (38.. ~) is equivalcllt to the relation

:!/I'I'.. _I(.z:) = x'J'..(x)

+ Y'~(.t)

("0.')

nnd equation (38.5) is unaltered in form so that 2.r'JI.. (x) = 211'1'.._I(x)

+ '1'"",,(.z:).

(40.5)

Eliminnting 2ulJ' ,,_I (x) from equntions (40A) lind ('la.S) we havc the rtlntiOll

From the point of view of mathematical physics the most imJlortant properties of Hermite functions concern integrals ill\-oh'inl-: products of two of them. In estnblishing llIost of these propcrties the sturting point is the observntion thnt the function 'P .. (z) satisfies the rclntioll (·'0.7)

138 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § ~O

+

as is obvious merely by substituting 2n 1 for ). in equation (.10.1). Writing down the corresponding relation for 111"" y"J:': (2m I - rx2)'P", = 0 (40.8)

+

+

multiplying it by y"Jn nnd subtracting it from equation ('10.7) multiplied by y"J,~ we obtain, as a result of integrating over (- 00, co), the relation 2(1I1-1I)J

•

•

'PmljJ~dx=S (t/J",lp~'_':P",y"J:~)dx. _00

_
Noll' an integration by parts shows that the right hand side of this equution hils the value n [ Y"J" ,ljI'

-

J' _S" (IP' 'P' _

ljJn'jl' '"

-~

",n

-~

lp'n'1" m ) d.

and, if we remember that, for nil positiw: integers 11, ljJ n(x) __ 0 flS lar 1-:1> 00, we sec thllt this has the value zero. lienee if we let

we sec that 1 m ... = 0, if

1/1

'*

'II.

(.10.9)

In particular I n_ l

,n+1

= 0

so that from equation (40.5) we have

J'-." 2x!Jln(x) IJln_l(rx)(/x =

211[n_I, .._I·

(-\0.10)

Now if in equation ('\0.3) we substitute for 11,,(x) from equation (38.2) we hll.\'e

d"

lj'{x)=(-I)"d:'-(e-=') n

d;r:~

(40.11)

\40

,,'

THE FUNCTIONS Of HERMITE AND LAGUERRE

so that the left-hnnd side of equation (-10.10) is equal to a>

S

-

d" dn - I 2xl:'" - (e- z ') - - (e-:t')d,'l:

_00

dX"-1

11.T;"

und nn integration by part... shows LhaL this is equal to

+

I"." 1"+1,,,_1 Le. to J", n' Hence rrom equation (.10.10) we have J n. PI = '211/ .._ 1, n-I'

Hcpcating this operation '0,0=

11

1imcs and noting' that

r"-.

c-""dx= ,In

we find that I n.n = ""/1 •t \!; .,

Combining cqul1liollS (.\0.9) and (·10.12) we have finully (.10.13)

The c\'fduntion of more complicutcd integrals can be effected by combining this reslllt; wilh the n:CIllTCIlCC formulnc we 11lWC nlrcndy estahlished for the I Termite fUlIctions. For instance. it follo\\'s from equalion (.10.5) that

I:,.;1:'l 1 (X) lJ.I,,(X)dX = m

Ill m ,n_l

+ t/m,n+!

showing thllt

I~ a/Pm(x)~fl,,(.l;)dx =

-.

und that

r'-.

0 if

x1fl,,(X)IP"+i(x)dx = 2"(11

#

n

±

1

(-IO_H)

+ I)! V~.

(.lO.I5)

11/

Similarly, mnking usc of equation (·IOA) lind (.'quutions

140 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 41

(40.13-15) wc ean show that

{tl

'.II m(X)lf'~(X)cJX =

-a>

0 if m :;6n ± 1 2"-1./11 v;. if lit = /I _ 2"(1l 1)1\1ii if

1

+

_

1

In

=

/I

+ 1.

Thc Occurrcncc of Hel'mite Functions in Wave 1\'1echanics. 'fhe Hermile fUllcliOlIS which \\'e Illl\'e discussed ill the last section ocellI' ill the wave Illecllllnicni tl'eallllellt of the harmonic oscilllttor I). Although this is II very simple mcchllniclIl system the analysis of its properties is of lp'cnt importlll1cc uec:allse of iL<; applicntion to the quantum theory of radiation. 'rhe Sc:hrodingcr equation corresponding to n harmonic oscillator of point lllass 111 with vibrn1.ional frequency)' is 41.

,Plj!

8n211l

-I.+-'~ (I

/ X'"

j~

I'

~

~~)

-:.?Jl~I/II'~x-V'=O,

(.B.I)

where II' is the total encrgy of the oscili:ltor ami h is Vlanck's constant. The problem is to determine the wavc functions 'P which havc the property that (i)

(ii)

'11-,0 liS

Ixl-+oo:

-.

fa>

1 tp 1211,'/;' = 1.

If we lct

then the eq\ll1lion

(.~1.1)

hecomes

,Fiji (211' ~) Ijl=O _+ de:: --~~ Ill'

(.u .2)

lind the conditions (i) and (ii) become I) N. F. MoU IIlld I. N. Sneddon, "'m.... ,Hull/wit.' till/I pljtalil)lu. (OxforU, \fIlS), p, 50.

1/11 .'1/),

141

THE FUNCTIONS OF HERMITE AND LAGUERRE

(;')

lp--+ OIlS

IE 1-+ 00;

f-.• I'P[:di =

(ii')

~7l

,fmo. f '+-,-,

The nrgulllcnt ghocli at the beginning of Section '10 shows that equation (-n.:!) possesses solutions 'P which satisfy the condition (ii') if and only if the parall1eter (211'11/1') wldeh OCCllrs in the equation lllkcs one of the \"3111C5 1 ~II where 11 is n positive integer. In olher words solutions of this type, which IIrc known by the probability interpretation of the w:wc function l;" to correspond to sial iOllllry stales of the :')"SIClll enn exist if nnd only if

+

II' = IIr(lI where

II

+!)

(41.3)

is a positi\"c integer. '''hen this is the case the form

of the wave [unction 'P is known from Section ·W to be 'P = ClJ1 ,,(;)

(.HAl

where C is It constant.. Applying condition (ii') unci equalion (.IO.I:!) we see thnt

C _ (.t-rml)I__'_ -

21"{1d)i

Ii

Thus the wlIve function corresponding to nn ndmissable energy {It ! )hl' is

+ ,

If.

~

(1:1:11/1)1 h

YJ,,{~)

::I"(I/!)I'

~=

'2,-r,r;;;; z VT

In quantum theory the matrix clements (1/ defined by t.he equntion (II

J:r 11)

=

-.

(.H.O)

I it: 11')

fa'> XV,,{.r)v'..(z)tl.T

fire of consiclernble importance in the elise of the hnrmonic oscillator. In terms of tile vnrinble ; wc hn\'e

(" 1'1,»

~

h f· <~.«)I'.l<)d" -,-.• apllll' _a'>

142 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

t

'2

so that substituting from equation (-n.O) we have

1]/-" --J•

;1JI.. (nlfl.,(';)l/;-;-{2!'*I;I(tll)l(pl)I}. rr .1:1: III)' _0<: It follows then from equations (<1o. H) and (.10.15) that

(I'lxlp)=-

(1I1a:11)=0

if1';P11±

nnd that

(1I1·TI"+1)=

I

+

1)11 .. 8n·1Il1'

(II

(111%1"- 1 )= { -HI, .&"t·IIII'

I

('1.7)

JI ,

(·11 .8)

JI .

42. The LaCuerre Polynomials. Thc Laguerre poly. nomials L .. (%) nrc defined fOI II n positive integer and % (l, posith-c reul lIumber by the equation

X') (1 - t ).,...... L,(%) - ,'.

(

exp - - - = 1-t

..-Oil!

Expanding the exponential fUlIetion we scc that

The coefficient of

l~

in this expansion is ;'(-1)'(r+I).._ , ~ '( ....0 r.1I r.

),'

Using the relations lit

(-I)"

(r+ I)..-,=;t, (I. we

.!iCC

that this

SUIll

(-1I)r

r)!=n!

..

can be written in the form )~ (- 11), • ~-( • -0 r.

')'

(·12.1)

I"

THE FUNCTIONS OF HERMITE AND LAGUERRE

,<3

,n

Idenlifying the coefficient of with D.(z)/n! and adopting the Ilotation of Section 11 we see Ihnl L,,(z) = 11! IF1(- II; 1; z) ('~2.2) It should be obscn re
erD"(x"r") =

II

(_II)

' (D"-·.z:")(D'c Z ) r! where D denotes the 01}Crnlor dJdx. Using the relations eZ/J'(e- Z) = (_ I )r, n"-'x" = 'l!z'/r! t:'" ~ ( - I)r .-0

we sec thllt c"J)n(x"c%) = II!

...." ~

(-,,) -,)/(x , y.

It follows immcdiately from cquation L,,(x)

(.~2.~)

,.

(.~:?:?)

,'"

= r dx" (x"c- Z ).

that (4:?A)

The first five Lngucrre polynomials enn be calculntcd ensily from this equation; we find thnt L(l(x) = -I, l~dx) = 1 - x, l,z(x) = 2 -
+ +

+

+,rI.

Equntions (.12A) can be used to show that thc functions I

9',,(z) = -I cl"L,,(z)

"

(4-Z.5)

form an orthonormal system. 'From (oIZ.4) we hnve as a

14/; THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 42

result of

III

integrations by parts

f"o r,-"x"'L.(x}(lx = f0

d",.. (3'''e~'')(I.:r: :J!" (X

oo

= (-1)"1II!

d.. -TIl - -TIl (x"c")d.r

fo dx"-

and this is zcro if It> Ill. Sincc l ...(x) is a polynomial of degrce III in x it follows that

I=r"L..(il')L ..(lE)dx =

0 if 1110:;6:

(42.6)

tl.

Since thc term of dcgree n in L .. (z) is (- 1 )"z" it follows that, thcn m = II.

J~ r"{L"(x)}!(t:e = =

(-1)"

J~ c-r;r"l.. ,,(x)dx

fo"11Ix"c-"(I;1:

= (111)2 Combining this result with equation (.l2.0) wc find that

I= c"9'..(x~..(x)llx = "...,

(42.7)

showing that the rp's form nn orthonormal set. Rceurrence formulne for the Laguerre polynominls may be deri\·cd directly from the definition (.l:!.l). Differentiating bolh sides of this cquution with rCllpcct to /. we obt.ain thc identity _ _"_ ex) (_ (1_/)2 . . 1

.E-) ~ (I_I) ~ I...(x)t"-I _ ....0(/1

1-/

1)1

~ L,,(x)t" .. _0 II!

which mil)' be wrillen in the form

x ~ £,,(%)'" ...-0

fI!

+ (1_/)' ~ L.(z)/"-1 _ ..-0('1

1)!

(I _ I) ~ £ .. (x)'" ...-0 ,,!

o.

§43

THE FUNCTiONS OF HERMITE AND LAGUERRE

Equating to zero the coefficient of In in the cxpansiolL 011 the left we obtain t,he recurrence relation /..,,+t(X)

-I- (.1: -

-I-

1 )l,,,(x)

21t -

11 21..,,_1(.1:) =

~ L,,(x) ,,_0

I"

-I-

~ r.~(x) /" =

(1 _ t)

II!

,,-
/1

(.12.8)

(.~'.!.l)

Similnrly if we differentiate bolh sides of respect to x we obtl1in the identity t

o.

with

0

I

which yields the Tetlln"CnCe relation L;,(x) -

td.~_dx)

-I-

IIL,,_t(x) = o.

Differentiating equation (.12.8) twice with respect to nnd rcplncing n by '/1 -I- 1 we find lhat L~'+~(x)+(x-21l-:J)f~:"+I(X)+(11+ I

oX

Fl.:: (x)+2L~+1 (x)=u. (.~2.1O)

Now from U2.9)

L;.. (.• ) ~ (n

+ n{L;(.• ) -

L,,(x)}.

and hence L~'+I(x)

= (II

-I-

1 ){L~'(II)x -

l.;,(x)}.

A similar expression for L:.'+d:r.) in terms of [;,,(.1:) :md its derivatives call be readily ohtaincd. Suhstituting t.hesc vfllIICS of L~+dx) lind l~~+I(x) in equlItion (·1,2.10) we find t.hat

43. LaguclTc'S Diffcrclllial "Equ:lIion. Equat.ion (.1:?8) shows that y = ALn(a:) is a solution of Lagucrrc's differcntial cquation

(ry

ely

(X-

(X

x-I~+(1-a:)-1 +l'y=O

pa.1)

in the case ill which v is a positive inlcgcrll. If we put

146 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

f

'3

Y = 1, <X = - " in equation (11.2) we see that it takes the form (,(3.1) so that it follows from equation (1104) that one solution of equation (.13.1) is Yl(Z) = tf'I(- "; 1; z),

('13.2)

Similarly we see from equation (11.8) that the second solution is Y2(x) = YI(IV). log x

+.-,•~ c,.xr

(43.3)

where the cocfficicnL~ cr lIrc defined by CfIlialion~ of t.he type (11.!J). Thc gCllernl solution of Laguerrc's differentinl cqulltion lIlay therefore be written in the form

Y = AYt(x)

+ By!(x)

where A and 8 are constnnts and YI(X), y!(x) nrc defined by equations (.13.~) alld (43.3) respccth·ely. 1f we nrc interested onlv in solutions which remain finite at z = 0 it is ob,;ous'from equation (-13.3) that we must take the constant 11 to be 7.cro. Vurther if 0. i.. the coefficient of r in the series expansion for YI(X) it is easily shown that 0r+t/o....... rl. As the result of 11 discussion similar \.0 that ndnlllced in Section :JO, it follows thnt if II is not nn integer

yd.r) ...... tr

IlS

x -)-

co.

If, thcrcfOI'C. we nrc looking for solutions whieh incrcnsc less rapidly than this we mllsttnke v to n positive integer, in which cllse Yl(X) reduces to n polynomial. The equlltion (.1:1,1) possesses II solution which inerctlses less rapidly limn e% as .r -+ co if nnd only if the parameter I' occurring it is n positive integer, 11 s..'lY. If it is also rcfJuired that the solntion shall remain finitc nt x = 0, the solution is of the form

Y = AL,,(.:r)

(.13..Q

whcre A is a constant and L .. (x) is the Laguerre polynomial of degree 'I.

THE FUNCTIONS OF HERMITE AND LAGUERRE

44. The Assoclalcd La~ucrrc Polynomials nnd Functions. Tr we differcntiAte Lliguerrc's differenlilll cqulllion III limes with respect to z we find that it becomes t/..+2 y It-+1 y d"'U x -+ I.. +(III+I-X)-1 + ,+(',-11I)-1-=0, lX'" lIlX'" M

whieh shows thot

I~:'(x)

defined by

"m

L:;'(x) = ~ I,,,(z). ,~m

is

1\

(II ~ III)

solution of the c1iffe.·ential eqlllllioll

Xd·.. +(III+I-X)-j· + (11-III)y=O. x'x fr-'J

tllJ

(.I.k~)

The polynomilll/..:'(x) definet! by C
(-1 )"'(,.1): . \P\(-II+III; 11I+1;%) (II~III). (.&1.3) m!(1l III)!

SimilArly equation (.12..t) lends to the formula 1..:;'(.1:)=

(l"'lr%~(.r"r-Z)l.

d.c'"

tl.r"

(.UA)

The simplest nssocintcd Laguerre polynomials nrc; l.\(x) = -

I.

I~(.rl = - ·1

+ 2.1:.

I";(.:r) = 2

l";(.z:) = -IS+ 1S.r-3r. I";(x) = IS-Gr. ~(I)= -6.

I~:(x) = -OaT 14IX-4&r-4r, I~~(z)= 111-90.1:+ 12r. L~(x) = - 96

+ 2·lr,

L~(z) = ::?I.

The definition (H.I) for the IlSsocilltcd Laguerre poly. nominl is the one usually tnken in Applied mnthemntics.

148 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 44

In pure mathematics the fUllction

+

- (III II)! I' ( L •,.,(") I I 1 1 III '1

. m

".

+ 1;

z)

(.U.5)

which is a solution of the differential equation rFy

"-/ . + (Ill + 1 (X"

,I,T; x) -/ (y

+

Ily

=

0

is often taken as the definition of the associated J~llgllcrrc polynomial I) so that cnre must be taken in rending the literature to ensure that the particular convention being followed is understood. It is readily shown from equation (.~:!.l) which defines the generating function for Lnguerre polynomials that the nssociatcdLngucrre polynomials may be defined by the equation (_ 1 ).,. exp (_

~) =

(1 _ 1)-+1

1-1

~ L:'~X) til. ('~.UJ)

.._mil.

This identity can Ihell be used La derive recurrence relations for the associated Laguerre polynomials similar to those or C(luations (4~.8) and (-i~.9) (er. Examples O(ii), (iii) below). The La~ucrrcfunctions 1l III(x)uredefined by thccCJuntioll RIII(m)

=

Id"'a:ll,~~l(;t)

(n ~ l

+ 1)

+

(.1 ~.7)

+

If in cqllntioll ('J,~.2) wc replnce 111 by 21 1, 11 by /I 1 llnd y by el"'WI!l we see tlmt t.he function H"I(x) is II solution of the ordinnry linenr differential eqllalioll

(4.1.8)

III most physicol problems ill whieh this equation IIriscs it is known that I is nn integer. By reasoning: similllr to I) See, (or iJlJilllllCf', fo;. T. Cop'Mln. F,mctiol" of 0 Compla (Odord Uni\"l~nil~' l'l'1'SJI. 1035) II. :!6D.

Voriobf~.

'"

THE FUNCTIONS OF HERMITE AND LAGUERRE

that outlincd in Section ·13 it cnn rcnclily be shown thnt thc equntioll (4 LS) possesses n solution which is finitc nt x = 0 nnd tcnds to zcro as:z: -)0 00 if. and only if, thc paramctcr II which occurs in it assumcs intcgral vnlllcs. Whcn this docs occur thc solution is i'/U ..I(x) whcrc the function U .. 1(:z:) is givcn by equation (....1.7) and A is an arbitrary constant. We shall now evaluate the integ-rot

1,,1= f:.r{R",(X)}2('X whieh nrises in wave mechanic.... From equation (·H.G) wc ho\'c thc idcntity ..

.,~

,,_7+1

:If

.,~

/21+1( )Z21+1( )

_-..+1

.x

~,,'+I

x 1,,+1 ,,'+1_

"'~+I (11+1)!{tl'+ 1)!

T

-

c.'\:pl-~-~} I-I t):!1+2(i

(1

I-T

r-I+I :!I+I

T)Zl+2

T

from which it follows that 00.. .,~ ')' ~ _

.... 1+1 ,,'_/+1

1"+1 .. '+1 T (+'),(' I)' 1/ • 1/ + .

T)":I+2

f0

..

_r "1l"J 21 +1 ( )1 2/+1 ( )1 M~"+I .'1: ~,,'+/:r IX

c;r-

f" "'+" I-,1:---- - 0

r

,TI 1-1

MCXp

'x

orT } I

1-7

This last integration is elementary and gi\'cs for thc cxpression Oil thc right (2f

+ 2)1(IT)":/ll(1 (I

-

l)(l -

T)

iT )11+3

and by mCllns of the binominl theorem we mny cxplllld this funclion in the form (I _ t _ T _ IT)

...i:

(21

+ r- + 2)! (IT):!l+r+l.

"

150 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY f'S

Now the codricicnt of (tr),*1 in this expansion is 211 {(u 1- l)!} (II I 1)1

+ 1)1}2. Ucnce " "{I''''I (Z »'dZ -- ,''«,,+'),)' ( I .

and this is C
fo

I.J{(1l

.:t'

"I)!

(·H.9)

45. The Wave FUllctions for the H}'dro~cn Alom. We shnlt conclude this chapter by disclissing the motion of n single electron of mass m and charge - e in the Coulomb field of force wilh potentinl JI{r)

Z<' = --, ,

due to a nucleus of c1mrgc Zc. In a first approximation we Illll)' Lrent the lllUSS of the nuclells os infinite lind this elISe the wave fUllction 'P of the s)'stem is governed by the SchrOdingcr equation I)

&~'J1I (II' + -z~ \1'=0 ,...,,+-h'! r rio!

(45.1)

lind the conditions:

+

1Jl(r, O. tp 2)1') = VJ(r, 0, 11') for nil r, 0, rp; 'P mllst be bounded in the range 0 &: 0 =::;; 3¥ for nil r, Pi (iii) 'P -+ 0 as r -+ 00; (i\') 'P remains finite as r -+ 0; (i) (ii)

(\.)

f

V' is normnlizcd to unity, i.e. 1\1 12 dT = I where the integrnl is tuken througho1lt the whole of space.

IV is the tot.'ll energy of the system. I) N. ).'. Mott alld I. X. Sneddon, loc. cit. p.

Ci·~_

THE FUNCTiONS OF HERMITE AND LAGUERRE

\45

15'

The cqul\tion (.15.1) may be solved by setting 'II = R(r)8(OJf/J{p),

(45.2)

where, by the method of scplirnlion of \'uriablcs we have

tPll' -~

(1",-

+

~

11-(/1

= 0,

(.15.3)

_._I_-"-(SillOdG)+{f(l+l)_ ,II: 0 dO (10 sm"O

Sill

18 =0,

~~(,dR)+{8n''''(II,+Z''')_I(I+l)IJl~ • I ' I I" O. r-{

(f

/'"n

r

/,-

(HA) (4:) n.•))

To satisfy condition (i) we mllst choose liS II solution of ('15.H) a function (I)(rp) such that (/J(rp 2Jt") = p{p) for all rp. Thus U OCCUlTing in eqllution (45.3) mllst be an integer und a convenient solution will be

+

(.15.6)

where A is nil nrbitrnry constant. Equation (.15.5) is the well-knowlI equal ion of which the solution is the IISsocintcd Legendre polynomial ll~(cos 0). If / is integral and /,~ I nl, then P;'(cos 0) is the only solution which is bounded in the range 0 S 0 S;T nnt! is therefore the only onc lcnding to 11 wa\"c function \'1 which ~atisfic~ condition (ii) abo\"C; if 1 is not an intcger 110 bounded solution cxists. To solve equation U5.5) we write a2

= _

811: 2/11 II'

112

'

I' -

-

·~n2I1lZe2

(l5 . . 7)

h2a

and change the imlcpcndcllt variablc from r

\.0 IV

whcre

:r: = 2cu. \Ve thnn find that

(12/.1+':" tiN _{-.!.._~+I(l+.1)1 dx~ :v d:v .~.e £"

fl= 0

('15.8)

where, in order that conditions (iii) and (iv) should he satisfied, R. must be such that U --)- 0, as x -)- 00 nnd as

152 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § d

-» O. From the flrgumcnls of Section 4·l it follows that this is possible only if I is a po!>iti\'c intcgcr nnd only if l' is lUI integcr, 11 say. which is greatcr limn 1+ 1. Whcn this is so wc mar writc thc solution of cqulltion (45.8) in the form (H.i) so that thc solutioll of (.15.5) is propor· tional to 1l",(2:u). Now by thc second of thc equation.. (45.7) we ha\'c

.J::

(45.9) for thc possiblc \'nlucs of thc towl cncrgy II'. The wm'c fuuctions corresponding to the \'nlue of encrgy given by the intcgcr II nrc ('l5.1O)

whcrc C nln is n constant determincd hy thc condition (v). III polnr coordinatcs dT = r" sin 0 drdO dp so thnt this condition givcs

1 = C;h. ('dip

I:sill o{l';' (cos 0WdO

$0 that it follows from cquations C

.I~

~ow

or.

I:

(~I.~O)

r2 U;,('.!lXr)dr

Ilnd (.l4.9) tlmt

'{(-1)"(I-U)I(u-I-I)!(21+1)}1 (- ) ·Ia.ll 8.Tll(I+II)I{(,,+')!}1

=21X ()2

from equations (-15.1) and (45.9) wc find thnt

= 4:r"Ze2mf(h"1l) so thnt if we introducc thc Bohr

radius a by means of thc cquation a~

/0' -·1·:r:me2

(.15.12)

we find UllLt or. = Z/all. Introdl.cing IIlis rcsult into (45.11) and substituting thc \'aluc obtnillcd for C,,'u into cquation (.15.10) we find that

I"

THE FUNCTIONS OF HERMITE AND LAGUERRE

153

'I',,,.. (T, 0• .,,) = 27.)3. (l-II)!(II-1-1)!(21+1)ji U (2«T)P~(cosO)e/"" {( fltl (l+Il)I{(II+1)1}3S:T11 .., (45.13)

with" $ I::;;; ,,- I arc the wave-fullctions corresponding to the energy (.IS.!l) of the hydrogen nlom. rr we write 11'11 = - 2:t':Z'!ef mlh': Ihell corresponding 1,0 the encrgy 11'0 we ho\'c the wnve fUJlction lJlll10(r, 0.",)

Z: c-P = V1lt (n)

lind to the energy level

lI'o/'~

(Q = Zrja),

we hn\'c the three wave

functions

1

Z •

'f'~(r. 0, Ip) = .~ V(e.."'l) (a) I

Z

I

(2 -

Q)e-h.

~

'f':IO(r. O. /p)

= -, II" ) (-),~, cos O. '"\ _:I:

'I':m(r, O. '1')

=

(I

1

Z •

ee- i , sin (} e:H ".

Sv'(:t) (;).

From the last of these fUlIctions we Cfill construct two fUllctions I

(7.)"I echo sIn. 0 sin 'P'

-r- .~ V ~Jf (l

cos

Similarly corresponding to the cucrgy 11'0/0 we h:\\"c the six W11\'C functions

Z)'

I

V'300(r. 0, 9') = S 1V (-' (2; 3:T(1

.")

Z·

'hlo(r, 0.1]') = - - - (-:,)" (0 8J,:Ta

V'311(r, 0,1]')

1 7.)'• (0 = -/(....: 8t, 7t (l

JSIl~

+ 2e~)fq,.

e)erf' oos O.

e)~I' sin 0 e H ',

154 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY §.o\5 'j/3:!ll(r,

O• tp) =

(Z):.(l-r I

1 8] vOn -;

'/'3,,\(r, 0,1]1) = 8 " ~ l' v n

q

(3 cos·• 0 -

I),

(z)te~c-hSinOCOSOe±j9', (/

'P3~(T, 0, 9') = --2-/ (~): I?~rh sin:l 0 C:£:21." • 102,\ 7l fl In WlI\'C mechanics IIp{r, O. tp) l~dT represenls the proonhilily thnllhc electron whose wave-function is II'{T, O. tp) is to be fOil lid in n small volume 1ft: centred lit the poinl wilosc poInt coordinates arc (T, 0, rp). To make the totnl prolmuility IInity we must 11ll\'c

f I '/I(r, 0, 9') 1 {IT = 2

I,

where the int.egral is taken throughout the whole space. Since, in point coordinates, (17: = ,.2 sin 0 drd 0 drp it follows tbnl the probnbilily Lllnt the electron is nt n distance between r - ~dr lind r ~dr from the electron is 91(r)dr

+

where

rp(r) = ,2 f:"'111}'

J:

sin 0 dO . I 'per, 0, tp)

I~·

Hcnce for nn clcl'tron in lhc statc dcfincd by qunntum numbcrs II, I, U (cquntion (45.1:J) Ilbove) wc havc 26)3 {II - 1- 1)1 ~ ~ tp(r)E:::: ( all :!1l{(11 lllP r-{R"I(2C(r)}~.

+

The lllcnu \'liluc of nny function I which dcpends on r alone is given by

1 = f~ I(r)rp(r)dr thul is, by the formula

( + Ill); f"" "/I'l{II,(2.'l)',I,.

.... X)3(Il_l_I)1

1 ~ ( =-lilt

2/1 (II

0

];'or examples of the use of lhis formula see example 17 helow.

THE FUNCTIONS OF HERMITE AND LAGUERRE

1SS

EXA11I'I.t:S I,

!'nH'e that it III

< n

.,.. (1I.(.r)} _

d.r-

2.

Ir

K(.r, y. I) -

:!-II t ---1/._(1). ,It)!

(,, -

•~

'i'.(zh.'.(y)l·

0-0

where the "".(;r) IIrC the orthonommlllCt o! Ilennite !,,"dioM detined b)' the relation prove \hl\t

f

•

'J'-I,'+oll.-l'" . e -I_·+ll._l',.( \.r.y,I...-_e

-0

A~slll1linll lhlll l\(x. y, /) is ot tl,e funn A I:Xp(lJx' + CJ:y where A, 11, C mill JJ arc rllncliO!l.~ of I pro\'c t.hut

N(.t.lI. I) .. 3.

";:;+::7.m •..•. ,{ .I.ryl - (.I"+y')(1 +1') ~("'{l 'I'll ~"'l :l(1 I)'

+ Oy')

I

SIlO", Ihut

o

,. (I/.(.r)}" _ _

~

"t

..-0

... Thc Schn'dinl-,'cr efllmtion tor the lhrce·t!imensiolllli oseilllllor ill C)'lintlri{'f\1 ('()()rdinatM (e, 'T, :) is 8:'1''''

1"'.,. + h! {II" -:l:'l'm(.'{I' + w1;:')}y -

O.

Show thut llnl IIOllItiollJl "" or thi!i equation ",eh that"" _ 0 as : _ cc IUitl ... i'l finite at Ihe oril:"iu ure ot the form

Cexp (iu

-I""{l' - ifl':'I/, •

where C is n constant, t, 11,1" I ure 11'05ith"c

(1- z."Vlmw/I.). alllll, .. IE; is

It

alld

(~lll.(;)

illtl'~t!I.

IlOl)'lUlrllinl

{l

«-';!.:tV(lIlr/h), ~l in; wllieh

ot degree

-'lttisfiCll the C(luatloll

"'f , ) "f ("1"1+"1-"'j - - (..:_~. -. E -+ ~ E' 1_0.

156 THE SPECIAL FUNCTIONS OF PHYSICS ANO CHEMISTRY

Sho\\' also that the corrcspmlllinu vfllucs of II' nrc t::;vcn hy thc Cllllfilion II' _ (2/ ju I + i)IH'+ (II + j)"w.

+

5.

Pro\'e thnt [,.(2.1:) .. III

III

2.-'"(-1)"

II

1:

m_OIll.

f,.

Ill.

Prove Ilml (i)

8.

)1 [, __.. (;1:).

fl

-(/.. , [,.(;1:) _ L.'H' (.1:)

,.

Prove l.hnt if m

'" J o

::i

'I, (/

'"

>

0,

(-1)"(111)'(11-1)·-" C-";t" I,. (;t) 1/.1: - '-=,!::-"'C';f:~-'-ftH

lind tieduce that if n

m)!fl

(II

~

I

+

I {(Il+l)I)I (_1)•• 1:;:11 .. (II I I)!

9.

Sho\\' thaI (1l!)~1

I..

J

lI''"(l-:r)·[•• (fl.J')IU _ (-i) •• '

o

iO.

{(11+1!)l)'

"

·'(11). '

I'rove Ihnt the function

xiI' c-j~ 1.:+,,(;t) iJ " solution or the differential equlIlion li"y ,,-)'+1 u}I ( lixl+--.--d.l:+

I

-"'4+

2p+,,+1 _ }'(2"-,) ) y _ O. ~ .1,1:'

THE FUNCTIONS OF HERMiTe AND LAGUERRE

157

II. The potential cnergy ror lhe nuclear molion or 0 dintomie molC'Cule ill c1Ol1e-I)' "I'llroxillllltetl h)' the )IOMle runction I'(r) _ Dc'" - '.!Ve-, Show ahut the spherically symlllctrical solutiOll.'! or the Schrlltlillger c'luntion with Illi~ I'0tcillini lire

C.

I

\'(r) ~ ~ {exp( - be--J}(~r)·-·- I.:,::~:~' (2be--)

(0.s;1/ :iA' -

H

where b _ :!.:l(~mV)! '(all), C. iJ II nonn:l.liZ3tion L'Onstant, and the \'Iduoo or the ene~' ore

oorn::spolltlin~

1I' __ lJ

1

[

(II

+ b

j)

+ (II + _HI) .u.'

12. Prove tl'ut the normLllimtiou constant C. in the "revi/l!ls eXnlnl'le Il:I~ the ,·:tlue

I'Illcre

13. Show lImt in pllrn\)olie L'Oonlin:ltlS e'llllitions Sdnvdinccr'~ Clll.llltion

~.

'I, Ip dcfim,
II)'

'he

ror a h)"
tllkes the ronn

-''/-) ". ~' SI,ow tlmt thi" e'luation hilS soluliolli of lin; furlll I{(;.

'I. '1') - c ( :1} I) cl.... ':. ~II'''I.; ... (.!.) I.~ .• (,,-) no 110

nil

where n i:il the 1I0hr flulit,ll ••, _k t-l+" j-l and 11'_ -:!,,·l/Ic·:·/(u·,,'J. Dctennine thc L'Ollstunt C w IIml \~ b nonn"li:tetl tl> 'IIlity.

lSS THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY 14. In the theory l,f the mtntion nnd \'ihrntion "llCCtrul11 of" dl". tomic molecule tlll'l'e nri.st'll the prolJlem of soh'ing the Sehmdins::et equatioll with potential Clll'J1lY II 7.,' l'{t) - -;:1- -,-. Clll'tf(~' l..,·cls ure s::iwn by II' _ - 2",'IIlZ','f("'rr') II + I + V{v + (I +I)'} with II, I integer.; und v_ lIud tlllIl 11'e cotrcspon,lillg wa\"c functions nrc

Show tliat the where

a_

871"11111/1,',

C ••

(2r)i"-I'_""~f."

~.a

(Jr'

(2t) P;(eosO)d~'I' (JII

where" is the lIol" tullius, a. _ 2,/{b -I- (I -I- H'}uud C." lllJllisntion fllclor.

i~

II 1I0t·

15. Show UU1\. the eonstllilt C•• oc('urring ill the !:LsI exanlJlle the \"uluc I) 16.

}

:{

(2/+1)//!(1-.1)! 4:r(211+a.+1)(l+u)!

IUI~

)!

If

1(/: II:

prove tllUt 1(1; expllnsion or

II;

8)

+

.J - J~ ;roH{Il.,{.l:)}"llz

+ I)l}' is

{{tl

(I-I)'''(l-T)'tl

U,e coefficient of {Ir)·" ill the

t

(21 +" +

r

r!

t

+ 2)! (IT)='totl.

Deduce t1w\. 1(/;11;-1)

I (I: II; I)

{(II -I- I)!)" (II

Ill!

{(u -I- I)')" {G,,' - 2/(1 , .)' {II t .

+

Il)

17. SllOw tlUI\. ill II J,ydrogen·like ntom of lIuclenr chnrge Z, the ll\'eT'l\ge distuncc of tl,e electron fmlll the HlLcicus, in the ~tntc described hy (Illllntmll Il11l11hcJ'!l I, II, ;!I

"'" [ I 7. l\

+:l, {

I -

'i)]

III + --,,-,-

Find the IIVl'nlA"C value of I{r lind ~ho'" thllt the total energy or hplrogell "tom i.'l just o"r-half of the ll'"ernge potent;ul ellerl:Y.

,\I'I'~:SUIX

THE DIRAC DELTA FUNCTION The Dirac Delta Function. Tn mathemnticnl physics we often Cllcollllter fUllctions which havc lion-zero vllllles in very short inten-als. For exam pic, an impulsive force is envisaged liS acting for only II very short intervnl of lime. The Dirac delta function, which is llsed extcnsively in quantum mcchanics and classical npplied mathematics, may he thought of us a gcncralisation of this concepl. If we consider the function 46.

d.(x)

~ {,'a' 0,

[z I <

(Ii

Ixl >

a.

(40.1)

then it is rcadily shown that

f'-"

ll,,(x)dx = I.

(40.2)

Also, if I(x) is un)' function which is integrable in the interval (- (1,0) thCII, uy \ISing the menu vllIIlC theorem of the integrul cnleu]lls, we sec lhat

S"

1 _,.,/(:r~,,(z)tlz = 2(1

S·_/(z)tlx = I(Oa), 10 I,;;

I. (.'0.:1)

We now definc

.....

6(z) = lim 6,,(.1:).

(.10.4)

Letting a tend to 7.ero in equations (.W.I) nlld (40.2) we sec Ihat d(z) satisfies the relations ll(x) = 0, if z =1= 0, lilll

(.Ul.S)

160 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY' 46

r-.

6(.)d.r

~

1

(46.6)

The "funetion" 6(.), defined by equations (4G.a) lind (40.G), is known as the Dirnc delta function. It is unlike the futlcLions we normRlIy encounter in Illnthell1utics; the hiller Rre defined 10 Il:\ve 1I definite vnlne (or \'nlllcs) nt each poin\. of n certnin dOlllnin. For this rCllson Dirac has cnlled the delta function llll "impropcr function" lind hns cmphasised that it may bc used in mathematical analysis only when 110 inconsistency call possibly mise from its lise. The delta fUllction could be dispensed with entirely by using a limiting procedure im'oh'ing ordinary functions of the kind 6.(z), but the "function" 6(x) and its "derivllli\'l$" play such a useful role in UIC formulation and solution of boundnry value problems in classical malhelUuLicll1 physics ll." well liS in qUllutum mechanics that it is irnporllllll to derive the formal properties of the Dirac delta function. It should be empllllsiseu, howevcr, I.hnt these properties lire purely fornlal. First of nIl it should be observed that the precise vllriation of d (.x) in the neighbourhood of the origin is not important provided lIll1t its oscillations, if it has any. arc 1I0t too violent. For instance, the function

. ) ,. ~,;:::n.':('::~::'::""1 u(z=lma-.'"

;"IX

siltisfies the e'luntiolls (.10.5) tunl (.10.0) and has the same formnl properties us the funct.ion defined by equation (olGA).

If we let t.he relation

(I

tend to zero ill equl\Lion

(.~o.a)

r-.

we oblain

1(·}
(....7)

whieh n simple e1illllge of variable transforms to

f'"

-.

I(z)d(x - ali/x = I(a).

(....8)

THE DIRAC DELTA FUNCTI ONS I" 161 In olher words the operati on of llIultiplying I(x) by o(x-n)

lind integra ting' o\'cr nil x is merely equiva lent to substitutin g a for z in the originn l fundio n. Symbo licaJly

we may write

I(.}
~

f(a}
(4G.9)

if we remem ber tlml this equatio n hlls n meAnin g only in the sense thnt iL<; two sides give cquivn lcnt results when used ns faclols in an integra nd. As n special case we have xo(x) =

In a similar way we

o.

Ctlll Jlron:~

(4G.I0)

the relation s

b(- x) - b(.), o(ax) = -

1

o(x),

u

(J>

Let us now conside r upon the "dcrivn tivcs" exists alld lhat both it dinary functio ns in the sec lhat

0,

(40.1~)

the interpr etation we must put of o(.t). If we nSSUnlC that o'(.r.) lind o(z) can be regarde d uS orrule for integra ting by parts we

ncpcnt ing this process wc find that

Jao f(x)l5I"I(X)dx =

-.

(_ 1 )";1"1(0),

The statem ent is often made thnt the Diracd clta functio n is the derivat ive of the Hcn"is idc unit functio n lI(x) defined uy the equatio ns

JI(x)={6:

ir if

x> 0;

x< 0;

and it is cnsy to sec on gcolllctricul ground s that thc,'c arc

162 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY §

~6

rClIsons fm' conjecturing such a relationship. To mllke it precise we note thnt if, in the definition I) of the Sticltjcs int.egrlll

r-.

/(0)"1'(0).

we tnke F(z) to be the lIcfi\'iside function /-I(x), we find immedifitcly thnt. for lUlY illtcgrnblc function I(:c),

r-.

/(oldll(ol

~ /(0).

(40.15)

Compnring cfluntioll (·lG.l5) with cqunlion ('lQ.7) we see t.he relation between //(x) nnd 15(x). It may be secn from thesc equat.ions thllt 15(01') is not n function but II S!iclljes measure, lind Lilli\. the usc of the Dirac delta function could be avoided entircly by II s)!stemn\.ic IISC of Sliclljes integ:rat.ion. ') TIIC

fiimplest dcfillitioll of tl SticlljCll illlegmi f:J(J:)dF(:r) is

liS

the lintit of nJlproxinmtj"c sums ~J(e,) [F(:r,) - 1'(:r,_,}], where lht' z, nrc l'0inl.ll ofsubdj"ision of (a, b) I1ml ;,/ies in thc inlerml (:r,_" :r,).

INDEX ,\,hliti..,,, formula for 1l6Se1 coefficientll 101, 1:10, ,\itlwn, A, C. 13. Anl\lytic fUllction 5, AIHlm"ly ilL. ASNUl·i:ltccl I,:lgllcffe fUllctions 1.17: POI.\'IlOmillls 1.17. ,\~yrnptotic cXJlansion~ o{ BeMcl (unctions I:!'~,

J)ir"e
IInile)·. W. X. :17. Uei {unciions 11-:-. lIer funclioM 117 lIebel <'OCfficicn15 0:1: {ullctions ,I; intcj.'l'1I1 function Ial. n I's differentinl equutionJi 4. 07. 102. Ucla (unction 10. Uohr rodiu.'! 152.

G'U1ulla fllnction 10. Gauss' Ihcorem ~I. Gellemli",," h)·JICfl,'C.'OlIletrie series :16. Gillespie, H. I'. 10. GOIlrs.'lt, E. a. Gm)', ,\. I:! I.

Cuueliy's tht'Ofcm 70. Circul:1f fUIlCliollll -I, 115. Completc ~e'lllellee liO. Cornprc.,,~ihlc

1I11nkel'lI lIClisd fllllclion~ 107, ll;lfluollic t>.'lC.ilI"tor lolli, 155. Ilcll\'iside'" unit fUlictioll 101. Ilt:rlllilc functions lau; poly· llominls l:l:!. IIl'rmilc'li diffefCntial C
flow:! t.

COlliluclion of helll. c1luution ·I:!. COllfhu:ul. 1I)'l'crgt:ullll'l de eflulltion a:!; JlypcrJ;couU'lric func_ tion it!. C<>ntiJ:uit)' relationJl :II, :l5. COll$On, Eo T. 148. CO!iine·inll~l.'J':ll 12, 17, Coulomb field 43, Uti. ('ouhon. C. A. 71. C.m..ilineaf eo-ordinnla I, la. C)'lilldril'lll L'O·ordinnleof :1, 113, I:.!I,

l:U.

Ilobsoll, Eo "'. 503, 5Q. SO. lI)'dro~'C1i IltOIll laO, 157". 158. 1I)-JlCl"Jtellllletrie CfI""tiOIi 2:1; fUlletiollJl 18: series 18. lI)'slop, ,I. :\1. 18.

H.a.

Inee, E. I.. 5.

Indicial "fl"ation ,. Intef,,'I'll1 funllllillc 20, :la, 7U, I{)/l,

I.)eltn function 150. llinlomic molccule t57, US,

110.

'"

164

THE SPECIAL fUNCTIONS Of PHYSICS AND CHEMISTRY

Jnhnke, E. 14.

ll~lIrrence

Kummer's rel:ttion :15, Laguerre's diffCn'lItinl ~Iuatjoll 140. J..ngllcrre I'olyllomluls J.I~. LaJilace's C(lulltion I, 1.1, 15,46, 70, 83, 00,

1~1.

Lcgendre fllnelion~ 0:1: ]lol~'­ llomials -10; IICrlCll 5S. l..eg<:lldre's Il.!l~(lcililcd crllUIUOIl 7:1: u,!I!;ocialeert .... M. ll!, 00, I~~. )lathcW!l, G. II. I:!~. "'Intrllc dementll III. )lclA'lchlan. :". W. 110. .'IOIltfif'd Bessel f,mclion, 113. .'Ioleclllllr sl~tn& :!~. 157_ :'Ilorse function 15•. :'11011, ~. F. I,W, l1iO. :'Ilurph)"s fonlluln 51.

relation 6; rel:tlions for l.egendre funclions 5:!, (:{I; relntioll!J for lJ.euc:l funelion~

.

,

llqllllar sinJ;1I1:1T point G. Iliemnnn !'-function :!5. lloderiAucs' formula 55. lIolltlor sa. lIulhcrfortl, D. E. I, 01. Sn:tl~cllnt:;:'~ theorem 41. !::icllri~dingcr cquillion 2·~.

'1'01.11:5 of function!! 13, H, 51,

fllllCliulI~

5ll.

Ordillltr)' point G. OrUr0l,>OlIal &et"1"CIlL'e 59. Orlhonormal IICt GO. Parnbolic co-onlillfttes 15, 157, l'uulillg, L., 2.1. P-funclion :!5. !'hiJliI)!!. K G. 70. Point III infinity O. Potcntial UlCOr)' '10, 57, ,0, 83, 121.

1~-1.

Tnylor 5eriC!! 5.

F. K 115.

:"eum:ulII's adLlition lhcorcm 130 Bessel funeliOIl 105: formuln

"", N'orrrllllisctl

.~3,

Serie.. of Legendre function!! 57: or Bessel flll1eljolL~ II U. !::iillc·jntcgral I:.!, 17. !::iinJluhtr IKllnt II. Snedtlon, I. :-:. t·IO, 1110. !::il'cci:tl funelions 3. Sl'herie:tl Hessel functions lOS: oo-ordinate!l I·S, ':"0. !::itieltjel inlegml 162. ~urf:lI:e .r;pheric~ll h:muonics 80. S)'metrical-toll molecule 2'~, ·12. 110,

~eurmUIll,

,I:.!,

Ka. 140, 1[,0, Hi5, 157.

VllndcrlllOlltle'~ tl,t'(lrem ~2. 'VHI.~(lIl, G. l';. 1:!1. "",llIOIl'S ULL'(lrcr1\ 45. WIIVC mechnnics ,I:!, 43, 8:1, 1.10,

t,m.

Wclier';j Bessel function. tOn. Whil1llle's theorem 45. \\'llillllker'!1 fUllction, as. "'ilion, f;. B. 21. ...~fO!II

of

of U1.'endrc function. S,' frmcllons I:.!O.

I~l