Complex Variables with Applications - The Solution Manual

COMPLEX VARIABLES WITH APPLICATIONS Third Edition By A. David Wunsch University of Massachusetts Lowell SOLUTIONS MANUA...

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COMPLEX VARIABLES WITH APPLICATIONS Third Edition By A. David Wunsch University of Massachusetts Lowell

SOLUTIONS MANUAL By A. David Wunsch

8r. Michael F. Brown

Copyright 2004 Pearson Education Inc. Publishing as Pearson Addison-Wesley, 75 Arlington Street, Boston, MA 02116

COMPLEX VARIABLES WITH APPLICAnONS Third Edition By A. David Wunsch University of Massachusetts Lowell

SOLUTIONS MANUAL By A. David Wunsch

Michael F. Brown

Copyright 2004 Pearson Education Inc. Publishing as Pearson Addison-Wesley, 75 Arlington Street, Boston, MA 02116

The bulky volume you are holding represents the solutions to all the problems in the 3'd editioo of my textbook Complex Variables with Applications. Both Michael Brown and I have separately worked through all the solutions, but I can say with overwhelming confidence that in spite of this redundancy there are some remaining errors. Please tell me of any that you find. My email address is [email protected] (note the underscore). Those preferring an older medium of communication may write to me at the Electrical and Computer Engineering Dept. University of Massachusetts Lowell, Lowell, MA 01854. I promise to acknowledge all e-mail and postal mail that I receive. I would also appreciate learning of any errors in the textbook itself. I plan to post corrections to both the book and this manual at the web address http://faculty.uml.edu/awunsch/Wunsch_Complex_Variables/ This manual has been written primarily for college faculty who are teaching from my text. Whether it is to be made freely available to students- perhaps at the school library- is a matter I leave up to each individual instructor. Notice however, that there is little point in assigning the textbook problems involving computer programming if students already have the MATLAB code supplied in this manual. Regarding this code, I must assert that I am not a professional programmer and I'm certain that in many cases the reader will fmd more efficient ways of solving the same problem. Finally, I must apologize for the idiosyncrasies of the handwriting. They are my own and not to blamed on Mr. Brown. A. David Wunsch

Belmont, Massachusetts July 8, 2004

e'• +1=0

TABLE OF CONTENTS 1 Complex Numbers 1.1 1.2 1.3

1.4 1.5

Introduction More Properties ofCcmplex Numbers Complex Nwnbers and the Argand Plane Integer and Fractional Powers of Complex Numbers Points, Sets, Loci, and Regions in the Complex Plane

2 The Complex Function and its Derivative 2.1

In~ction

2.2 2.3 2.4 2.5

Limits and Continuity The Complex Derivative The Derivative and Analyticity Harmonic Functions Some Physical Applications ofHannonic Functions

2.6

3 The Basic Transcendental Functions 3.1 3.2

The Exponential Function Trigonometric Functions

3.3 3.4

Hyperbolic Functions The Logarithmic Function

3.5 3.6 3. 7 8.8

Analyticity of the Logarithmic Function Complex Exponentials Inverse Trigonometric and Hyperbolic Functions More on Branch CUts and Branch Points Appendix: Phasors

4 Integration in the Complex Plane 4.1

Introduction to Line Integration

4.2 4.3

Complex Line Integration Contour Integration and Green's Theorem Path Independence, Indefinite Integrals, Fundamental Theorem of Calculw in the Complex Plane The Cauchy Integral Formula and Its Extension Some Applications of the Cauchy Integral Formula Introduction to Dirichlet Problems- The Poisson Integral Formula for the Circle and Half Plane Appendix: Green's Theorem in the Plane

4.4 4.5

4.6 8.8

s Infinite Series Involving a Complex Variable 5.1 5.2 5.3 5.4

Introduction and Review of Real Series Complex Sequences and Convergence of Complex Series Uniform Convergence of Series Power Series and Taylor Series

5.5 5.6 5.7

8.8

Techniques for Obtaining Taylor Series Expansions Laurent Series Properties of Analytic Functions Related to Taylor Series: Isolation of Zeros, Analytic Continuation, Zeta Function, Reflection The z Transformation Appendix: Fractals and the Mandclbrot Set

6 Residues and Their Use in Integration 6.1 6.2

6.3 6A 6.5 6.6

6. 7 6.8

6.9 6.10 6.11 6.12

Introduction, and Definition of the Residue Isolated Singularities Finding the Residue Evaluation of Real Integrals with Residue Calculus, I Evaluation oflnt:egials, II Evaluation of Integrals, ill Integrals Involving Indeoted Contours Contour Integrations Involving Branch Points and Branch Cuts Residue Calculus Applied to Fourier Transforms ThoHilbertTnmsfonn Uniform Convergence of Integrals and the Gamma Function Principle of the Argument

7 Laplace Transforms and Stability of Systems 7.1

7.2 7.3 7.4

Laplace Transforms and Their Inversion Stability -An Introductioo Tho Nyquist Stability Criterioo Generalized FWlCtions, Laplace Transforms, and Stability

8 Conformal Mapping and Some of Its Applications 8.1 8.2

8.3 8.4 8.5 8.6

8.7 8.8

Introduction The Conformal Property One-to-One Mappings and Mappings of Regions The Bilinear Transformation Conformal Mapping and Boundary Value Problems More on Boundary Value Problems-Streamlines as Boundaries Boundary Value: Problems with Sources The Scbwarz-Cbristoffel Transformation Appendix: The Stream Function and Capacitance

9 Advanced Topics In Infinite Series and Products 9.1 9.2 9.3

9.4

The Use of Residues to Sum Certain Numerical Series Partial Fraction Expansions of Functions with an Infinite Number ofPoles Introduction to Infinite Products Expanding Functions in Infinite Products

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I f.7o ~ sfrc..t'lltl'li:£ YI"ArK•" 'f 1111 - "Vo [~:'t ;- rrx.J : 4>--!'I>OJ+ , Cl>"< Vol f ' 4>--~Ja. q ~o . StrcA~\tr>f-' l h -VOX: jf wrt£. a.rrows .U.< -1 ~~~~ \ .-. -.."{oltt- ... \ ' --- - -. -. ',·4- C..IIJ<"ctt< •~' .,_ 0 Arrows .r-r.t r~~ID/11 " (C.) ~(1!-.o~· o L'.f ., Lo. l tu.L-<-1. y Tn [,;.~t- : ,(, :.o o( .... •• :ur o<"3=0 X. 1), -.-JC. u.. .,_ 0 £,1 3 • "" A'w w"- /(. . . A ..to5 ( w•-~e·) + 13 wh•n A'.,. ~A ~e c 1 ("') )e_f- 1: 2 -- Alo~ l wz.- 1e •) +13 u,"t'•l. A-=- Ar+L ·-Chr . ·~ 1\r\ k_tV--0, Thus Ax, B:. lo~lw"-lc;~J (A.-;.0, Ar--o, C.cmt' e( $. 5 Br+L t (,(-..DO. Br ~- B:r Br+-~ 13r 1(...'\)IA.I~S Z!;....,.oo-(. 1 Asst,nHm+ Of br""'h; tala , ..t.r So rk'l u,i v> tiol.~ "';tl. l.o~-< t Lt.low) ___;<;..."-.;,;~"_-t-_Y_ (IA. t- -t~l<.~ / -....,_.;:~;.....--!-- _.JL ---- -- )(. t w~ IG [1-e tr~]''"'- (I~ fOS. tl1.r?li'\ArY ff y .. o, f.- .. 0. TJ.ui t/.it lo.Ot: ,·, I"'•Pp«< t"-~'0 V-f'P"" half o-f w I" lion~. 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'il. s- -t II>~ ' q "' .L. a-t' Ill ' IT h W.MAtV C.0Yl.o( 1Tjbll)s the. , E,--_1 s (J,)' Co>\~ e( q, t- ~ lj/ "' L - ,.!:.__ Lo ~ w rr [e (" tt e -IT'/'= to., w , {> ir -the a.d·u .. l h1i.~cs 1- Ltf tc. d·uivG bOIA. ru.{ .. t I (S.I ~ -;. - l- [ [ elrr e.'rr4>e·"-~-' t -rj.,. It t + i" ~ p IVf A n0 S lA o.IIA WW~.sf tk w/ l)t>lUI'S 1"''1)UI~MC tif Lo., w + I J. L~. Ke'rr~ e-rrj +~ tcr¢> -rr'f' tT y -· -· t;J -o 1 ii L'l.e-nf(( S(IT-t';,, tJ::1_'f'tif c.or~JoltJ. Q, (_(.) bt'.l\'3 ~ --1 ~-1-/3 ~t, ~11'\..W.t'l t'II.A~t t.~ +he. L1l"'~"vt- OV\ wh,J.- 2J? -:. 0 \<>IAII"' Y-=-o wh·<-h t's ~alfw 4 t \oc:'ll-1/tC.Ii' {'~ '"2.. f> V:\rs . ?.>11 ~ '( 11 Me o\-Y' ~ f k_ .t"'l""' p or\'"< I'\t-Co..Lr to rrt~ rOil Mns ~ \'l<"~...tt i'-e: v..t L$'.f +k• o~ vo~A.")et t'IIHt 1-1-t ~IMfoTei'I~S ~ +i-t h,11.VoYis M.•rro~" \l"lt.t'lei fN c..orrt'S\"OIIIAIVi~ .fi-e lXAI"'j'kJ o-f tO~I'blvL (~ S Sl,....ioJIII ,;.., "iboVe) ~ ~-1~~: ..,...hr.ts o.l~Cl e><\,i},i!'· l'\1rror IMa~ • ~'r-4V1o\..<", '-·':!· ti.. ~hAVIOil~ of \{-'-:. C '/<0 MIA~\' \x -tl-.. t'11rror ll"'a~ O-f +k ~......du·ie $AIIA.ei 0\1\ wh·'c.lr. ~ "" (' '/"'? 0. P-78" 9 Advanced Topics in Infinite Series and Products I ~ ! ().~ n '" 2"" ~~~· ... I(M .. _..o \ (,., "V n +o. f'aSS +o (J;.; t- l i,,",-\: - ""..1."'-~-- lAM<~ 1!""- (ll'Ol) + .L ., 2.o."'" m•. Cot>, (Tr"') - I - ().. "" 0 ' sv.I'\I'IPI.ttli>\ Sl-:>1'1 . wJ, l""l s i11J;11.._ - I ..,. I,,;, rrau..A( IT '3 ur-JI.. ( n... ) s,../, err.,_) ,. rr 3 _ n'' -1- .,_tr ~ en.,.) t 2" rr"s i;;;;& en-..) cr.1r - 6 A-10 o 411408\ Ctt"l "I'>Prl L•ltopl-/oll'r t& a,;A;,.,. 'Sec '1.1, )'rob 1 clear % p=oblern 1 (d) in section 9.1 a=li~space(.0001,20,100); z=a; nmax=length ( z); seriesm=lO; while (seriesrn<=50) for kk=l:nmax s=O; for n=l:seriesm; s=l/(n~2+z(kk)A2)+s; end ya(kk)~pi*pi/6; sumz (kk)=s; end plot I z, I sumz) ) ; seriesm=seriesm+40; hold on end yl=cosh(pi*z) ./sinh(pi*z); yl=pi*z.*yl-1; y~yl./(2*z.A2); plot(z,y,'r',z,ya ,'-');grid on xlabel('a'); ylabel('function and sum of series') title('problem 1, chap 9, part d') ch. q. p.'l.- ~'I, sec 'f, I problem 1, chap 9, part d 1.8.----,-----,-----,-----,-----,-----.----.-----.-----.----. , _______ ___ _ 1.6 . .. . . ·>······ .................,.... . ......,............• ........... , ... . 1.4 ·; II) '6 -----··· ·---; I 1.2 1 ..... ~ ~ 0.8 ------'----····-----~---- --------"····---------~-- g ~ , ....... ...... .2 0.6 ___________ _, ····~········ ---------------------- ··i-··----- na. t.>thrr..; -• ······· ..... . -,. =-::-t;:--+---·· :Z.p!"" 0.4 2 4 6 8 10 • d •. ,, f,J 12 14 16 18 20 t-=- t[~] \12.. c~-i][~W\[~}· s·~ r~1 ..... ~ [Vf]-~[~l] - -2 V\ :. - c::c t n~o N~w -"'{,U11.-t"e rc"l Then V0t, ,._.,.;. t"''.o:!l. -:.- ~ f'es -. -Ti [ I ~ t :: l'co.t-+.s. _ e•ch al.vl!le - 2 __,, ___....,. sm(tt e) -\- ,;.. 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(.f X.'> I+ e (f T-r: h =1 .2-=l \-( n" 2,----.---.----.----,----,----r---.----.----, ti 1.8 -------:--·············>········ -----·-··· 1" ···········+-- 5 1.e _ : : _________ \ -··;··- ----------r---~ --··----~-- a.. ~1.4 ...... L; ~ 12 ----'\L . . . . . . .. 1; ~ 1 ~ c~ 0.8 ~os 0.4 : ............. ; ·············•: f·········· ---:---- ! -------------~---------------:-------- .infi. MB ~-~-~-~~7.:~.·... 1... 1..""' .....:...............·,' ............. / s··;·.....................................,. T\V/- ················~// • • o.21L---;2;-----;,:---c4---;,---!.---:,---!,---J.:,-----;',, NUMBER OF TERMS IN PARTIAL PRODUCT 'S,<.C. 'l. 3 %PROS 10 SEC 9. 3 clear n=lO; \ zz=.9*exp(i*pi/4); pl1)~1+zz;vl1)~1; for k=2: n pI k I~ I 1+zz.' I 2' I k-1 I I I . *pI k-1 I ; vlk)~k; end plot (v, real (p) , v, imag {p) , v, real (p), 'o', v, irnag {p) , '*' ) ; grid; x1abel I 'NUMBER OF TERHS IN PARTIAL PRODUCT' I y1abeli'REAL,IMAGINARY PARTS OF NTH PARTIAL PRODUCT') text (7 .5, 1.25, 'imaginary part'); text ( 6. 2, . 7, 'real part' ) infinite_product=l./(1-zz) textl5.1,1.5,'infinite product~ 0.6768 + l.1846i'); p ..!.!\ f,,:.r- IMw.{, 5l..w: 1-ti!+!-' . . . ~2.~-1 1- ~ , .. Sec 'i'.J J 2(o.l to" a. n L~OH) we. h.t.l Ve : 3 Lo:>(.\-Ta~'l + a, .. . '.?, - "1-'d.n"" ·'· 1-a.n (a"'l<1 o \to;! ~tao1 _ ~ ~ I '* -,. ~;;- •~~ ~ l~- ~n-~ +~ ... ~ ~ (~/ ..... '2. T- J ~nt' t-IOnj\ ~+ .foli~>W, .+<.,, +r ta. vi::ll e l.Y\ ~'1/IA<~~ l •tY . J 3 La'"t'.. 1 ~ tan/ .m., 2:"-'-. 3 r. 3C -s,ec. 'f .3 prob :I-f ~a~\<. i , LO.>'l t t lfl.,\'" t CJ:JY\ 'rl J.. 11-, rea\\ tal() -- Lan t s .. I \-la,t lcAI'll -Ia.,[ <> 1 1\;~A s - usn1':l r- .s:\A. \t ~ C"'l l Lo':!On(Lt-tt~) -1\ ~ (J. -tk I'J •/..... y;c---1 ..:I:JrJVe. ; pi'otlk.l •/2 Y\ 1!/IOUO h /•r'Je +l...t lol• I~ •/ .... 19oib.J Wt. kk W -- L o'!) (I tan) UVI rclA~ l>YWit~OOH ~~· l:%:t> ().f) c•r~l-t: 1ft! L-"7\ ""'_5"' ' 2: no1:e. ' IW ti~ j.V IS [WI :. ~ .$ I w- •I ; I 1'2.- I YYI'-" sf-.. Wf-<1'\ /s mi" wh•"' W -=·I~ ~~pt~n) ~ 3/~ (f) proh. /%. <1. 3, $ lo~ Ct-tan) Tf' CD >1"0 ·~ II c.(. ~~ .,_ a" I Lo., (It an) 1 ~ '2..!fo.nr Now by Tku~ Tkw• .. by . !Lo., (.ItA~) -1-h¢ C.:.liV<'l!ltn(e -+t.. .......... "I[~ n=l1 Lo;~"") \ t '1-f . ::-= wl'l.l1'-bo11 .. ~o La~ C.OM~o.rt.sol"\ -h!st 2... !-. .st C.OI!porl'•~ .1 t C.Ol4V"'!>en ::E" n. -:..o ~f.ficl<"l0 {. i's C\ S,jiAMp 1;,~.1 I <>.(. . g.,, ru.l Y.,u<+ be ::&!' [Q.ll ( ;:€ ta.~ 11=Z> 'S<'J<J r J c:o J1 "'")" ~c. " t- . IS -bhe "=~bsolw+c .fo-t .. 0. l..o>\VC•!>•'1VC.. ~,...,.., (J t«ll) I =: ~ 1 ., a.~~ev!)es, ~?> \Lo,(t-ta~))~l~~ I , . +!.." ~v -tk [Lo~ (1'~-"11JI d.•vcr.,es Tku~ +1.. c.o~ve•~<>'lce O{ :k;~ (a~{ 1S Gl 1'\ece-rr c..ntli~ (or +k ~ kollff: :ll ~=o ~.to.: \ Lo?(t+an)j -l'k ""lh .f-• TJ\11 s -tl.o k•b~< ,r-~~.~.t;*r: l~•J n ..,.Ill w\lerc ~"'~'~ Col'\~ . .""" tJ * \l.o,(tta~)\,"';P-•1 suc.k tk.l I anl.(.•/s r'l:>N tuts . o.n $ttlctl~ ·~ 11 • -\:r>A• "'- ' n:>ll!. ~olli'"''' ~l.fs 11 1'\dt' ~ pro~}cff. -e 'A. ~. 1 L""(lt4n)l is . CD"V;.,•., [fer ,.. :>Ill ~K.se .. Ia.~ I ..,.v . N 011 I ;~6'$1/.ni) .•. -:!.,. • l L.,., (I HI\)\ ~ ta~ l t.c~~.:. VIIU s!•11 ,'.( n -::.11 £D 1'1\l •"'- -z"' " •• • -< (nrr-rrtJ foliDioliit4 l'\li\a~-~fl.r t~~ ~ +~fo !::o t -- I 11..... of ,;;., cl·1.,;, '! · Z. _9..0- ~'t _ ' :S~~ 'e Ll: ~<~' is aM~ E';)"" ~i! -•1-e . " ,J; se' w \"o ks (;) d tt2.:rr,+ ... :ttrr, 1.. ur /)~nj f1;jJ-.r~- L,J.Icr ,~ I a'\ "j , :2. .. -z+i. rr Ti...a11 o-f, : 1- - ( i f!- t2.Tr t- I 2 •£Lrr · · sec 1o -:JsvrJ,.~ [ e 9 .-t, pro), 3 (b), l,_ 1 1o~ (1-t i:.)-~-).o~ TT .,_. /()':) ll t L' Tre ..~ ho-b>, -6 iJe.s c.... ] Lawlr111.1AeJ {'+ .i. \t 'liT"') -t .••. 'IP as eJ nrt> of. e s~?: ::: e ,~,(It a~;,...) t lo~( If z· 1-.w~· -= s ,..,...h:: i! ~ )o,(lti"/fi}'··· ::0 (!~ ~:) (tt (t~ 1~~) .... (H .,~;rr') (t-t-i~/CitlT'i) (t+ t'"/('"'3) · · . %solves prob 4 sec 9.4 clear; p{ll~4/3;wlll~pi/2; for k~2:100; PI kl ~pI k-1 I* I 2*kl '2/ I I 2* k-1 1* 12* k+ 11 1 • ' w{kl~pi/2; xlkl~k; end plot(x,p,x,w);grid; text(1,1.58,'pi/2') percent_error~100*{pi/2-pi100I OYl I 1-•..L. .............,.. ............ ..,.............. ..,.. ... . 1.6 1.5 ... 1.~ 1.4 1.35 ••••.•••.•• T • ·:- ····:·············+············:·········· ------)·-··········T·······-----+-·-·········L .. . ········-····· ...........•.. .......... .:. ............,... . ·······-···t·· .\ ............; .. . .. ··> ·····-~ . . ...........'.... ··-····+······--·--·+ ·······----< ············<··· Number of terms n in partial produd p.t3 "oo if ,. No\IJ -;;E K= ~nt>) t'\1< W•~h "' £.oMf4fiSOI< j<. ~ (t'ltl) l..c>'Ver~eS. +-< -s -t) : (._VI prfV{. J -:=-"-_r_-==,--x-iJ.!] ..._ r .... n+l ~ u..-- ... tol'te.r~e'"' -1-1 I +l.ii 'G .•. ~.. ch. 'I %solves prob 8 sec 9.4 clear all hold off z=linspace(0,5,100); pi2= (pi/2); p={l-z.A2/pi2~2J; n=9 for k=2:n p~p.*{l-z.A2/((2*k-l)*pi2)A2) if k==S ; plot (z, p); axis ( [-. 01,4. 9, -3, 3]) ;hold on; end if k==7 ; plot(z,p);axis{[-.01,4.9,-3,3]);hold on; end if k==9; plot(z,p);axis{[-.01,4.9,-3,3]); hold on; end end z4=cos (z); plot(z,z4) ;grid on; cosz :. . "A . . . of .....,.. ···"'········· -1 If . . . . . . . . .. ::r·I P'•PU···· 0 n=( \ 0.5 . .~ ·~ . I n=5 ' 1.5 Pm ' 2 U m uP 2.5 ' 3 P P 'm 3.5 ' 4 Pmm ' 4.5 prob 8 sec 9.4 f-4'7 P j Sec 1 . ~' (:e) s vnlt ~ ... :5 '-"1 '!. " BL~) <:JOJ\ i! Q e~ o a.o e - b.t resul.~e ~~ [,~ ....... ~. ~tlr'IC. p .. rt" ~::0 ~ •+ Sl~p[~ i< F('tl'~-" aNo(;; ""i' J,;, Laur. Sen;,s .._,~ J o...J. F £-.,) o;:o p~-\hri~ 0(,~0\ s~' oot~,, st.¥-, U.:r> .. w.~~.,.., ·~ Poles U:>s h'!: - C8l'i! ., 0 ~c <. ~ t a nrrW 1-.!< w• '5. VI~ e\' qj ~ t S • n<s T~ ar<. wh. r<.. a.>"e soiPI• poles. i!: ':l.flll . fl"U po I• 'itnl<. f'\u.t" ...~p•,) o VI f' W 1.\l,. t. l 't =!d. n1!"' ,·r i.s f['fj \'\ (\'\'A 1)- J..f. , ). )I( 'l...PD; ' t....m.~r c.,.. CBO'!: exc..l~ .fo, nrt~,.-Lc 4. c aM " • as i li !A" r\'J F (l>)~ of i< .:il'l•l•11't rt u (L'"';, ,.:.. ., ~ <• ... "-i\.P«l'lsiQi-\ 5~~-I'SI->1'!: -~ ~i - - " • ........ o ht [r, f/11A:e t-51Me7 t e •rr --- ,/ ' '' All .• f'D/e, s bw trt-l\ )'<,. I . ,. (II VI -Hv.t ro[(...... ' .'~Ill{ ~ (1\t.f+E) We ~ be. .. MtJ ""' «" nn tiT[.,_ ·1111 ----1( "' "" r< 11'\lof· 0 { l't4 c.. b,,. ,J Yl ("lot -e,,yin J.. P-L/6 ~ro~lt11 tl, Lo>'\tiXVIeJ.; kJ..f>.J 2-V/-l F (~) {;) #... _, [:c CrJ-1'1 J~vrt>re oJr 1'~ j<'al~\:5. '] poles l.totMS< Rtl.s 't , f ('tJ :lt'lll' [IH · ' 1'1 "-0,1,2 .. r's tAVI~litrC. 5vMH!>IM~ Q SIMPle pol" ~ ~ - (.0<)"1. • ~ s~~-rslm<= = 1 "'" ~s F (e) S ~'l-t S""'"- NoW c...pp\1 h1t\a'J- kf kr. ?olcs. l'n~l-~t -tk o~ s, '11'\"e-trtet.llY lo~otph- lou-l>t~l1.S >~e-tS~-~ ('\ -;:;. I to 11 -\,o,.r pltteJ . Ito< c..~t ••• z- b~o-J a"- -.. o - .._ - ol'l ~+<. p~ev•ot(< -rJ... ~u1 +k+ +k'flt'O..._ ot tht,,.;. .fJ S C.n~l~lloll-b'o'" q .4 ll.N.l t.ondi1S/tll1: 1; 0 ., £~• -UPe1J-::;. J. L' 1J "'L bol>J.. -::0 S ikNo(; II= I "I~ CfY\ A j,>l ~~ Gt~ l L "'(J!lT):J ftc.., t; i!' c-&.r- coc -: 1 )'ll th 0

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