Complex Analysis (Mathematics)

Transcription

1 SHIVAJI UNIVERSITY, KOLHAPUR CENTRE FOR DISTANCE EDUCATION Coplex Alysis (Mthetics) For M Sc-I

2 Copyright Prescribed for Registrr, Shivji Uiversity, Kolhpur (Mhrshtr) First Editio 9 M Sc Prt-I All rights reserved, No prt of this wor y be reproduced i y for by ieogrphy or y other es without perissio i writig fro the Shivji Uiversity, Kolhpur (MS) Copies : 5 Published by: Dr D T Shire I/c Registrr, Shivji Uiversity, Kolhpur-46 4 Prited by : Shri A S Me, I/c Superitedet, Shivji Uiversity Press, Kolhpur-46 4 Cover Desig by : Prti Pritig Services, Kolhpur ISBN Further ifortio bout the Cetre for Distce Eductio & Shivji Uiversity y be obtied fro the Uiversity Office t Vidygr, Kolhpur-46 4, Idi This teril hs bee produced with the developetl grt fro DEC-IGNOU, New Delhi (ii)

3 Cetre for Distce Eductio Shivji Uiversity, Kolhpur Prof (Dr) A A Dge Ag Vice-Chcellor, Shivji Uiversity, Kolhpur EXPERT COMMITTEE Dr D T Shire I/c Registrr, Shivji Uiversity, Kolhpur ADVISORY COMMITTEE Prof (Dr) A A Dge Ag Vice-Chcellor, Shivji Uiversity, Kolhpur Dr B M Hirder Cotroller of Exitio Shivji Uiversity, Kolhpur Dr (St) Vsti Rs, De, Fculty of Socil Scieces, Shivji Uiversity, Kolhpur Prof (Dr) V S Ptil, De, Fculty of Coerce, Shivji Uiversity, Kolhpur Dr T B Jgtp, De, Fculty of Sciece, Shivji Uiversity, Kolhpur Dr K N Sgle, De, Fculty of Eductio, Shivji Uiversity, Kolhpur Prof (Dr) S A Bri Director, Distce Eductio, Kuvepu Uiversity, Krt Dr D T Shire I/c Registrr, Shivji Uiversity, Kolhpur Shri B S Ptil Fice d Accouts Officer, Shivji Uiversity, Kolhpur Prof (Dr) U B Bhoite, Ll Bhdur Shstri Mrg, Bhrti Vidypeeth, Pue Prof (Dr) A N Joshi, Director, School of Eductio, Y C M O U Nshi Shri J R Jdhv, De, Fculty of Arts & Fie Arts, Shivji Uiversity, Kolhpur Prof Dr (St) Ci Yeole (Meber Secretry) Director, Cetre for Distce Eductio, Shivji Uiversity, Kolhpur B O S MEMBERS OF MATHEMATICS Chir- Prof S R Bhosle PDVP Mhvidyly, Tsgo, Dist Sgli Dr M S Choudhri Hed, Dept of Mthetics, Shivji Uiversity, Kolhpur Dr H T Dide Krveer Bhuro Ptil College, Uru-Islpur, Tl Wlw, Dist Sgli Dr T B Jgtp Yshwtro Chv Istitute of Sciece, Str Shri L B Jle Krish Mhvidyly, Rethre B, Krd, Dist Str Dr A D Lohde Yshvtro Chv Wr Mhvidyly, Wrgr Prof S P Ptr Vived College, Kolhpur Prof V P Rthod Dept of Mthetics, Gulbrg Uiversity, Gulbrg, (Krt Stte) Prof S S Bechlli Dept of Mthetics,Krt Uiversity, Dhrwd Shri Stosh Pwr,, 'A' Wrd Sdshiv Jdhv, Housig Society, Rdhgri Rod, Kolhpur-46 (iii)

4 Cetre for Distce Eductio Shivji Uiversity, Kolhpur Coplex Alysis Writig Te Dr S R Chudhri Dept of Mthetics, Shivji Uiversity, Kolhpur (Mhrshtr) Dr U H Ni Deprtet of Mthetics, Shivji Uiversity, Kolhpur (Mhrshtr) Editor Dr S R Chudhri Deprtet of Mthetics, Shivji Uiversity, Kolhpur (Mhrshtr) Dr U H Ni Deprtet of Mthetics, Shivji Uiversity, Kolhpur (Mhrshtr) (iv)

5 Prefce The Shivji Uiversity, Kolhpur hs estblished the Distce Eductio Cetre for exterl studets fro the yer 7-8, with the gol tht, those studets who re ot ble to coplete their studies regulrly, due to uvoidble circustces, they ust be ivolved i the i stre by pperig exterlly The cetre is tryig hrd to provide otes to those spirts by etrustig the ts to experts i the subjects to prepre the Self Istructiol Mteril (SIM) Tody we re extreely hppy to preset boo o Coplex Alysis for M Sc Mthetics studets s SIM prepred by us The SIM is prepred strictly ccordig to syllbus d we hope tht the expositio of the teril i the boo will eet the eeds of ll studets This boo itroduces the studets the ost iterestig d beutiful lysis vi Coplex Alysis As tter of fct Coplex Alysis is hrd lysis, but it is truly beutiful Alysis The first topic is itroductio to Coplex lysis The secod uit dels with Mobius trsfortios The third uit itroduces the reder to the otio of coplex itegrtio Fudetl theore of lgebr d xiu odulus theore re the results covered i the uit four Uit five d six cover cocept of widig uber, Cuchy's itegrl theore, Ope ppig theore d Gourst theore Luret series developet, Residue theore with its pplictio to evlutio of Rel itegrls, Rouche's theore d Mxiu Modulus theore re the results cotied i lst two uits We owe deep sese of grtitude to the Ag Vice-Chcellor Dr A A Dge who hs give ipetus to go hed with bitious projects lie the preset oe Dr S R Chudhri d Dr U H Ni hve to be profusely thed for the ovtio for they hve poured to prepre the SIM o Coplex Alysis (MSc Mthetics) We lso th Professor M S Chudhry, Hed of the Deprtet of Mthetics, Shivji Uiversity, Kolhpur, Director of Distce Eductio Mode Dr Mrs Ci Yeole d Deputy Director, Shri Sjy Rtprhi for their help d ee iterest i copletio of the SIM Ths re lso due to Mr Girish Shele who hd te pis i typig the uscript d Mr Schi Kd for providig pritig copy of the uscript etly d correctly Prof S R Bhosle Chir BOS i Mthetics Shivji Uiversity, Kolhpur-464 (v)

6 M Sc (Mthetics) Coplex Alysis Cotets Uit- : Coplex Nubers Uit- : Mobius Trsfortios 6 Uit-3 : Coplex Itegrtio 5 Uit-4 : Fudetl Theore of Algebr d 4 Mxiu Modulus Theore Uit-5 : Widig Nubers d Cuchys Itegrl Theore 54 Uit-6 : Ope Mppig Theore d Gourst Theore 7 Uit-7 : Luret Series Developet d Residue Theore 78 Uit-8 : Rouche's Theore d Mxiu Modulus Theore 97 (vii)

7 UNIT - I COMPLEX NUMBERS Itroductio We ow tht i the rel uber syste, the equtio x + hs o solutio This leds to itroductio of coplex uber syste i which equtios of the for x +, where >, hve solutios This chpter itroduces coplex ubers, their represettio d bsic properties Defiitio The coplex ubers c be defied s pir of rel ubers {( x, y) : x, y } Equipped with dditio ( x, y) + (, b) ( x+, y+ b) d ultiplictio beig defied s ( x, y)(, b) ( x yb, xb+ y) Oe reso to believe tht the defiitios of these biry opertios re good is tht is extesio of, i the sese tht the coplex ubers of the for ( x,) behve just lie rel ubers; tht is, ( x,) + (,) ( x+,) d ( x,)(,) ( x,) So we c thi of the rel ubers beig ebedded i s those coplex ubers whose secod coordite is ero The followig bsic results sttes the lgebric structure tht we estblished with our defiitios Its proof is strightforwrd but evertheless good exercise Couttive lw for dditio : Associtive lw for dditio : Additive idetity : There is coplex uber ' such tht + for ll coplex uber The uber is ordered pir (, ) 4 Additive iverse : For y coplex uber there is coplex uber such tht (,) + The uber is ( x, y) 5 Couttive lw for ultiplictio : 6 Associtive lw for ultiplictio : 3 3

8 7 Multiplictive idetity : There is coplex uber ' such tht coplex uber The uber ' is ordered pir (, ) ' for ll 8 Multiplictive iverse : For y o-ero coplex uber there is coplex uber such tht (,) 9 The distributive lw : The uber x y, x + y x + y is If we write x for the coplex uber ( x,) This ppig x ( x,) isoorphis of ito so we y cosider s subset of x y x + y x + y x+ iy If we put i (,), the (, ) (,) (, ) (,) (, )(,) defies field Let x+ iy, x, y, the x d y re clled the rel d igiry prts of d deote this by x Re, y I If x, the coplex uber is clled purely igiry d if y, the is rel Note tht ero is the oly uber which is t oce rel d purely igiry Two coplex ubers re equl iff they hve the se rel prt d the se igiry prt Coplex Ple or Argd ple : The uber (, ) uique poit ( x, y ) i the ple The ple x y x+ iy c be idetified with the represetig the coplex ubers is clled the coplex ple The x-xis is lso clled the rel xis d the y-xis is clled the igiry xis Defiitio Let x+ iy, x, y the the coplex uber x of d is deoted by iy is clled the cojugte

9 Followig re the bsic properties of cojugtes + Re d I i is rel iff if 6 Defiitio 3 Let x+ iy, x, y the odulus or bsolute vlue of is o-egtive rel uber deoted by d is give by betwee the origi d the poit ( x, y ) Followig re the bsic properties of Modulus x + y The uber is the distce 3 if x Re d I y 9 Let x+ iy, x+ iy the ( ) + ( ) ( ) + ( ) x x i y y x x y y poits (, ),(, ) x y x y Hece distce betwee the poits which is the distce betwee the d is give by 3

10 Polr represettio of coplex ubers Cosider the poit x+ iy i the coplex ple This poit hs polr coordites ( r, θ ) where Clerly x r cosθ d y r siθ Thus x iy r( cosθ i siθ) + + r x + y which is gitude of the coplex uber d θ ( udefied if ) is the gle betwee the positive rel xis d the lie seget fro to d is clled the rguet of, deoted by θ rg We ote tht the vlue of rguet of is ot uique If θ rg, the θ+ π, where is iteger is lso rg The vlue of rg tht lies i the rge π < θ π is clled the pricipl vlue of rg If, re y two o-ero coplex ubers the rg rg rg rg rg + 3 rg rg rg We shll siply stte De Moivre s Theore : For y rel uber, cos θ + isi θ is oe of the vlues of (cosθ + i si θ ) 4

11 th Roots of Coplex Nubers Let r( cosθ + i siθ) be o-ero coplex uber, the w ρ( cosϕ i siϕ) root of if w, where is positive iteger Therefore, ρ ( cosϕ+ isiϕ) r( cosθ + isiθ) Thus ( cos + i si ) r( cos + i si ) ρ ϕ ϕ θ θ ρ r d ϕ θ ρ r d + π, where is iteger θ+ π ϕ, where is iteger However, oly the vlues of,,,, ( ) hs distict th roots d they re give by θ + w r cos π θ+ isi π + + is th will give distict vlues of w Hece where,,,, ( ) Soe Topologicl spects Note tht is etric spce with respect to usul etric d(, ζ ) ζ By ope disc, we e the set { : < } d is deoted by ( ; ) closed disc, we e, { : } d is deoted by ( ; ) B Ad by B Further ulus is defied s the set { : r< < R} d is deoted by ( ; r, R ) The puctured dis of rdius cetered t is defied by, B( ; ) { } { : } < Defiitio 4 A subset G is ope if, for ech S, there is ε > such tht B( ; ε ) G The poit is sid to be iterior poit of the set S if there exists ε > such tht B( ; ε ) S Further, iterior of S, writte it S, is the set S I { G : G is ope d G S} The closure of S, deoted by S, is the set S I { F : F is closed d F S} The boudry of S, deoted by S d defied by S S ( X S) Further, subset S is dese if S 5

12 Defiitio 5 A etric spce ( X, d ) is coected if the oly subsets of X which re both ope d closed re X d the epty set Further, subset S spce ( S, d ) is coected X is coected if the etric Defiitio 6 If G is ope set i d f : G, the f is differetible t poit i G if li h f ( + h) f h exists It is deoted by f '( ) d clled derivtive of f t Defiitio 7 If f is differetible o G, the we defie f ': G If f ' is cotiuous the we sy tht f is cotiuously differetible Defiitio 8 A differetible fuctio such tht ech successive derivtive is gi differetible is clled ifiitely differetible Defiitio 9 A fuctio f : G is lytic if f is cotiuously differetible o G Power series Defiitio A series of the for clled power series bout Ex The geoetricl series where,,,, re costts, is is power series bout d for <, Theore For give power series defie uber R by li sup R, the ) If < R, the series coverges bsolutely b) If > R, the ter of series becoe ubouded d so series diverges c) If r R < <, the the series coverges uiforly o { : r} Moreover the uber R is the oly uber hvig properties () d (b) 6

13 Proof We y suppose tht ) If < R, the there is r such tht < r< R, sup, there is iteger N such tht < for ll N r < r for ll N > Thus by defiitio of liit r R < r for ll N Thus the series is doited by the series r Sice the geoetric series coverges for < r, the series r coverges bsolutely for ech < R b) Suppose > R, d choose r such tht > r> R, < Thus by defiitio of liit r R sup, there re ifiitely y itegers such tht > It follows tht r > r the series diverges for ll N d, sice > r these ters becoes ubouded d so c) If < r< R, choose ρ such tht r< ρ< R As i () we hve < ρ for ll N Thus if r, r ρ ρ r d < Hece, by Weierstrss M-test, the power ρ series coverges uiforly o r 7

14 Defiitio The circle R which icludes i its iterior < R, i which the power series coverges, is clled circle of covergece Rdius R of this circle is clled rdius of covergece of power series d i view of bove result it is give by li sup R Theore 3 If is give power series with rdius of covergece R, the R li if this liit exists + Proof We ssue tht d let α li Suppose tht < r< α d fid + iteger N such tht r < for ll N + Let B N r the r rr < r B, N+ N N N+ N+ N r rr < r < B N+ N+ N+ N+ N+ N+ Cotiuig this wy we get, r B for ll N The r B r r for ll N Sice < r we get tht Sice r is doited by coverget series d hece coverges < α ws rbitrry this gives tht α R Now if > r> α the < r + for ll N As bove we get N r B Nr for ll N This gives tht B r s Hece diverges d so R α Thus R α 8

15 Exple 4 Fid rdius of covergece for the series ) b)! c) ( ) ( i ) Solutio ) Here d, Therefore li sup li sup li sup R Thus rdius of covergece for the series is R Tht is the series coverges i whole coplex ple b) Here d!!, Therefore! R + li li li ( + )! ( + ) Thus rdius of covergece for the series is R! c) Here ( ) ( ) i d ( ), i Therefore R + li li ( + ) Thus rdius of covergece for the series ( ) ( ) i is R d circle of covergece is i 9

16 Theore 5 Let f hve rdius of covergece R> The ) For ech K the series + () hs rdius of covergece R b) The fuctio f is ifiitely differetible o B(, R ) d furtherore, give by the series () for ll K d < R c) For,! f Proof With o loss of geerlity ssue tht ( f ) is Therefore f () ) We first prove the result for K Tht is the power series se rdius of covergece Let g hve rdius of covergece ' Sice R is rdius of covergece of Now we hve to show tht R R ' f, log log li li log li li R where d li sup R ' li sup R hve Therefore li e Thus li sup li li sup li sup li sup Therefore the series d hve se rdius of covergece If < R ', we write + <

17 Tht is if < R ', is coverget Hece R R ' Also if < R, we write + < for Tht is if Thus R R ' < R, is coverget Hece R R ' Thus d hve se rdius of covergece Siilrly d ( ) hve se rdius of covergece R Therefore by ethod of iductio for y K the series + hs rdius of covergece R b) For < R, let S d Now fix poit w B( ; R) Let R so tht f S + R +, the there is < r< R such tht w < r< R δ > be such tht B( w; δ) B( ; r) Let B( wδ ; ) Cosider f f ( w) S S( w) R R ( w) g( w) S '( w) + S '( w) g( w) + w w w Ad (3) ( ) w R R ( w) + w w w w + + w+ + w + w + + { } + w + + w + w

18 + { } r + r r+ + r r + r + r Sice r < R, r coverges + Therefore for y >, there is iteger N > such tht r < wheever 3 + N Thus for N R R ( w) w < 3 (4) Sice li S '( w) g( w), there is iteger N > such tht S '( w) g( w) < (5) 3 wheever N Let x{ N, N } Sice S S( w) li S '( w), for give >, we choose δ > such tht w w S S( w) S '( w ) < w 3 (6) wheever < w < δ Thus for give >, there is δ > such tht wheever < w < δ f f ( w) g( w) < + + w Hece f is differetible d f '( w) g( w) for ll w B( ; R)

19 Tht is f ' Siilrly f '' ( ) f ''' ( )( ) 3 3 d so o ( ) f ( )( )( + ) c) Fro prt (b) we hve f (), f ''(), f '''(),, f ( ) () 3 Thus Hece for f! ( ) () f,! f Corollry 6 If the series hs rdius of covergece R> the, f is lytic i B(, R ) Proof By bove theore if hs rdius of covergece R>, the f is ifiitely differetible i B(, R ) Therefore f ', f '' exists i B(, R ) iplies tht f ' is cotiuous i B(, R ) Thus f is cotiuously differetible Hece f is lytic i B(, R ) Result 7 A doi G is coected iff its ope s well s closed subset is either epty org 3

20 Theore 8 If G is ope d coected d f : G is differetible with f '( ) for ll i G, the f is costt Proof Fix i G d let w f Let A { G : f w} Clerly A If A is ope s well s closed, the by coectedess of G, A G ( ie f is costt ) First we prove tht A - is ope Now for A Let If B( ; ) > be such tht B( ; ) we defie g :[,] G G by g( t) f [ t + ( t) ], t The [ + ( ) ] [ + ( ) ] g( t) g( s) f t t f s s t s t s [ + ( ) ] [ + ( ) ] [ + ( ) ] [ + ( ) ] [ t + ( t) ] [ s + ( s) ] t s f t t f s s t t s s [ + ( ) ] [ + ( ) ] [ t + ( t) ] [ s + ( s) ] f t t f s s ( ) [ + ( ) ] [ + ( ) ] [ ] [ ] g( t) g( s) f t t f s s li li ( ) t s t s t s t + t s + s [ ] g '( s) f ' s + ( s) ( ) ( ) Therefore g '( s ) for s iplies tht g is costt Hece g() g() iplies tht f f w Therefore A Thus if B( ; ) the A tht is B( ; ) A Thus A is ope We ow prove tht A - is closed Let be liit poit of A, the there is sequece { } i A such tht li Sice f is cotiuous, f f ( li ) li f li w w Hece A Thus A cotis ll its liit poits hece A is closed 4

21 EXERCISES ) Fid the rdius of covergece of the followigs ) b)! c) (3+ 4 i) d) + i e) f) 5

22 UNIT - II MÖBIUS TRANSFORMATIONS I this uit we study Möbius trsfortios d their properties We begi with bilier trsfortio + b Defiitio 9 A ppig of the for S c+ d trsfortio where bcd,,, is clled bilier or lier frctiol + b Defiitio A bilier trsfortio S c+ d p or Möbius trsfortio with d bc is clled Möbius Rers ) Möbius trsfortio is oe-oe d oto + b ) If S c+ d, the dw+ b S ( w) cw 3) If S d T re Möbius trsfortios the So T is lso Möbius trsfortio 4) S + ( Trsltio ) S ( Diltio/Mgifictio ) iθ S e ( Rottio ) S ( Iversio ) Theore If S is Möbius trsfortio the S is copositio of trsltio, diltio d iversio 6

23 + b Proof Let S c+ d b Cse Whe c the S + d d b Let S, S + d d The o with d bc be Möbius trsfortio b S S S( S ) S + S d d d Thus S So S Cse Whe c Let d bc d S +, S, S3, 4 c c S + c The S os os os S os o S ( S ) d S4oS3o S + c d S4o S3 S + c S S d + c 4o 3 S4 S3 d + c bc d S4 c d + c bc d + cc ( + d) c 7

24 Thus S S4oS3oSo S + b S c+ d Theore 3 Every Möbius trsfortio c hve t ost two fixed poits + b Proof Let S c+ d Let be fixed poit of S( ) the S with d bc be Möbius trsfortio + b c+ d + ( ) c d b which is qudrtic i Hece it c hve t ost two roots Therefore every Möbius trsfortio c hve t ost two fixed poits otherwise S for ll (Idetity p ) Theore 4 Möbius p is uiquely deteried by its ctio o y three distict poits i Proof Let,, 3 be three distict poits i Let Sd T be Möbius p such tht S w, S w, S( 3) w3 d T w, T w, T ( 3) w3 The T o S T S T w T o S T S T w T o S T S T w Thus T o S is Möbius p hvig three fixed poits Hece o ( Idetity p) Thus T S T S I Defiitio 5 For the p deoted by (,,, ) cross rtio if it ps, 3, 4 respectively to,, 3 4 where, 3, 4 is clled 8

25 More precisely the p (,,, ) to,, d is give by 3 4 is Möbius p tht ps, 3, S,,, Rers 6 ) (,,, ), (,,, ), (,,, ) ) (,,, ) respectively 3) Let M be y Möbius p such tht M ( w ), M ( w3 ), M ( w4 ) the M (, w, w, w ) 3 4 Theore 7 If, 3, 4 re distict poits d T is y Möbius p the (,,, ) ( T, T, T, T ) for y poit ( Cross-rtio is ivrit uder y Möbius p ) Proof Let S (,,, ) d T is y Möbius p Let 3 4 Now, S( ) M o T( ) M( T ( )) S( ) M o T ( ) M T ( ) S( ) M o T ( ) M T ( ) Thus M ( T,, T, T ) the S T ( T,, T, T ) Let T 3 4 the T Thus (,,, ) ( T, T, T, T ) o 3 4 Therefore S( ) ( T, T, T, T ) 3 4 M So T, the M o T S Theore 8 If, 3, 4 re distict poits i d, 3, 4 w w w re lso distict poits of, the is oe d oly oe Möbius p S such tht S w, S( 3) w3, S( 4) w4 Proof Let T (,,, ) d M (, w, w, w ) 3 4 Let 3 4 S( ) M o T ( ) M T ( ) M w S( ) M o T ( ) M T( ) M w S( ) M o T( ) M T( ) M w S M T o the 9

26 Thus we hve Möbius p S such tht S w, S( 3) w3, S( 4) w4 Uiqueess: Let R be other Möbius p such tht R w, R( 3) w3, R( 4) w4 The o ( ) R S R S R w R o S( ) R S( ) R w R o S( ) R S( ) R w Thus R o S hs three fixed poits, 3, 4 iplies tht o R S I Therefore, R S Exple 9 Evlute followig cross rtios ) ( 7 + i,,, ) b) (, i,,+ i) S,,, Sol We hve, 3 3,, 3, ) 3 3 Therefore, ( i ) b) ( i i) 7+ i 7 +,,, 7+ i ( i) i,,,+ + i ( + i) i + i i i i Exple 3 Fid Möbius p which ps the poits, 3 i, 4 oto w, w i, w respectively 3 4 Solutio Let S be the p tht tes i wi (,3,4) uder y Möbius p, (,,, ) ( S, S, S, S ) Therefore, (,,, ) ( ww,, w, w ), where Thus (,, i, ) ( w,, i, ) i Sice cross rtio is ivrit S w i i w i i + + w+ +

27 ( i) ( w i) 4 + i w+ i ( i) ( w i) i+ 4 i w iw+ i ( i)( w iw+ i) ( w i)( i+ i) 4 ( ( ) ( ) ) + ( )( ) ( + 4 ) ( + 4 ) w i i i i i w i i i i i ( i)( i) i( i+ 4 i) ( ( i) i( i) ) + i( i) i( 4 i) ( ) w i i i i+ 4 i i i + i+ i 4 i Therefore, ( i) 3 + i 3+ i w + 6 i+ 6 Theore 3 Let,, 3, 4 be four distict poits i, the (,,, ) is rel uber 3 4 iff ll four poits lie o the circle Proof Let S (,,, ) 3 4 the S is Möbius p fro theore we hve to prove tht { w : S( w) rel} is circle to To prove this Suppose S( w ) rel, the S( w) S( w) w+ b Let S( w) cw+ d with d bc Thus, w+ b w+ b cw+ d cw+ d Therefore, ( c c) w + ( d bc) w+ ( bc d ) w+ ( bd bd ) () Cse Whe c is rel Therefore, c c, the fro () we hve, ( d bc) w+ ( bc d ) w+ ( bd bd ) () Let α ( d bc), β ibd ( bd ) the () becoes,

28 α α β w+ w+ i ( α + α ) + β i w w ( αw) i ii + β ( αw) I β (3) Let α p+ iq, w x+ iy the α w px qy+ i( qx+ py) Therefore, ( w) I α β ( qx+ py) β Thus (3) represets lie q y x+ β p Tht is, w lies o the lie deteried by (3) for fixed α d β We ow tht stright lie y be regrded s circle with ifiite rdius Therefore, w lies o the circle Cse Whe c is ot rel Therefore, c c, the fro () we hve, ( d bc) ( bc d ) ( bd bd ) w + w+ w+ ( c c) ( c c) ( c c) d bc bd bd Let, δ c c c c Therefore, w + w+ w δ ww+ w+ w+ δ + ( w )( w ) + + δ + w+ δ + w+ δ + Therefore, w+ λ (4) d bc λ δ + > c c where Sice d λ re idepedet of w, (4) represets circle o which w lies

29 Theore 3 A Möbius trsfortio tes circles oto circles Proof Let S be Möbius trsfortio Let Γ be circle i d, 3, 4 re distict poits o Γ such tht S w, S( 3) w3, S( 4) w4 The w, w3, w 4 deterie circle Γ ' We cli tht S( Γ ) Γ ' : Sice cross rtio is ivrit uder y Möbius trsfortio, for y i (,,, ) ( S, S, S, S ) ( S, w, w, w ) 3 4 is rel Now Γ (,,, ) 3 4 S w w w is rel (,,, ) 3 4, S Γ ' Thus S( Γ ) Γ ' Theore 33 For y give circles Γd tht Γ ' i there is Möbius trsfortio T such T( Γ ) Γ ' Furtherore we c specify tht T tes y three poits o Γ oto y three poits of Γ ' If we do specify T( j) wj for j,3,4 (distict j i Γ ) the T is uique Proof Let, 3, 4 be distict poits o Γ d w, w3, w4 be poits o d M (, w, w, w ) S,,, 3 4 Let T M S 3 4 o, the T( ) M o S( ) M ( S( )) M w o ( ) T M S M S M w o T M S M S M w Thus T is Möbius trsfortio tht tes Γ oto Obviously Möbius trsfortio is uique Γ ' Γ ' Let 3

30 EXERCISES ) Fid fixed poits of diltio, trsltio d iversio o C ) If α+ β + b T ds Prove tht T S iff, + δ c+ d α λ β bλ cλ, δ dλ, for soe coplex uber λ 4

31 UNIT - III COMPLEX INTEGRATION I this sectio we shll study coplex itegrtios of coplex fuctios d estblished fudetl theore of clculus for lie itegrl We show tht lytic fuctio hs power series expsio s Tylor s theore For the we estblished Cuchy s estite to prove Cuchy s theore We begi with eleetry defiitios Defiitio A pth i regio G is cotiuous fuctio :[ b, ] G for soe itervl [ b, ] i If '( t) exists for ech t i [ b, ] d ':[ b, ] is cotiuous the is clled sooth pth Also is clled piecewise sooth if there is prtitio of [ b, ], t < t < < t b, such tht is sooth o ech subitervl [ tj, tj ], j Defiitio Let :[ b, ] G deoted by { } ie { } { ( t) : t [ b, ]} be pth the trce of is { ( t) : t [ b, ]} Note tht trce of pth is lwys copct d it is Defiitio 3 A fuctio :[ b, ], for [ b, ], is of bouded vritio if there is costt M > such tht for y prtitio P { t t t b} υ( ; P) ( t ) ( t ) M < < < of [ b, ] The totl vritio of, deoted by V( ) is defied s V { P P } υ( ) sup ; : prtitio of [,b] Defiitio 4 A pth :[ b, ] is rectifible if is fuctio of bouded vritio 5

32 Theore 5 Let :[ b, ] be of bouded vritio The: ) If P d Q re prtitios of [ b, ] d P Q the υ( ; P) υ( ; Q) b) If σ :[ b, ] is lso of bouded vritio d α, β the α + βσ is of bouded vritio d V( α βσ) αv βv( σ) + + Theore 6 If :[ b, ] is piecewise sooth the is of bouded vritio d V b t dt ' Proof Firstly we ssue tht is sooth so tht ' is cotiuous Let { t b} P t < t < < the υ ; P ( t ) ( t ) '( t) dt t t t t b '( t) dt '( t) dt Hece, V ' b t dt, so tht is of bouded vritio Sice ' is cotiuous it is uiforly cotiuous Thus for give > we c choose δ > such tht '( s) '( t) < wheever s t < δ Also we choose δ > such tht if { < < < t b} d P t t b t '( t) dt '( τ )( t t ) < where t b t '( t) dt + '( τ ) t ( t t ) { } P x t t : < δ the τ is y poit i [ t, t ] + '( τ ) dt t t 6

33 t + ' ' + '( t) dt If P δ i( δ, δ) b t [ τ t ] dt t t < the '( τ ) '( t) < for t i [ t, t ] d '( t) dt + b + ( t ) ( t ) Lettig ( b ) ( P) + +υ ; + b + V + we get '( t) dt V Thus V ' b t dt b Theore 7 Let f d g be cotiuous fuctios o [ b, ] d let d σ be fuctios of bouded vritio o [ b, ] The for y sclr α d β : b b b ) αf + βg d α fd + β gd b b b fd α + βσ α fd + β gdσ b) Theore 8 If is piecewise sooth d f :[ b, ] is cotiuous the b b f d f ( t) '( t) dt Proof To prove this theore we cosider rel d igiry prts of, we reduce the proof to the cse where ([ b, ]) { t b} P t < t < < hs P < δ the b t f d f ( τ ) ( t) ( t ) < d t For y > choose δ > such tht if 7

34 b t f ( t) '( t) dt f ( τ ) '( τ )( t t ) < for y choice of t τ i [ t, t ] Now by Me Vlue Theore f ( τ ) ( t) ( t ) f ( τ ) '( τ )( t t ) for soe τ i [ t, t ] Cobiig this with bove two iequlities we get b b f d f ( t) '( t) dt < Sice > ws rbitrry we hve, f d f ( t) '( t) dt b b Defiitio 9 If :[ b, ] is rectifible pth d f is fuctio defied d cotiuous o the trce of the (lie) itegrl of f log is f ( ( t) ) d( t) lso deoted by f f d b This lie itegrl is Defiitio Let :[ b, ] d σ :[ cd, ] be rectifible pths The pth σ is equivlet to if there is cotiuous fuctio ϕ :[ cd, ] [ b, ], which is strictly icresig, d with ϕ( c), ϕ( d) b, such tht σ o ϕ The ide is to recogie ll the pths hvig se trce s ideticl The bove defiitio brigs bout prtitio of the clss of ll pths Thus we re propted to defie A curve is equivlece clss of pths The trce of curve is the trce of y oe of its ebers A curve is sooth (piecewise sooth) if d oly if oe of its represettive is sooth (piecewise sooth) A curve C is clled siple if it does ot cross over itself Tht is ( t ) ( t ) wheever t t A curve C is clled siple closed curve if i) ( b) ii) ( t ) ( t ) curve wheever t t, except whe t d t b It is lso clled Jord 8

35 Theore Let be rectifible curve d suppose tht f is fuctio cotiuous o { } The: ; ) f f ; b) f f d V ( )sup f : { } c) If c the f d f ( cd ) + c We shll coclude this chpter with Theore Let G be ope set i d let be rectifible pth i G with iitil d ed poits α d β respectively If f : G is cotiuous fuctio with priitive F : G, the f F( β ) F( α) We ow prove Leibit theore: Theore 3 Let ϕ :[ b, ] [ cd, ] be cotiuous fuctio Defie g :[ cd, ] by b g( t) ϕ( st, ) ds, t [ cd, ] The, i) g is cotiuous fuctio d ii) If ϕ exists d cotiuous, the g is cotiuously differetible t Moreover, g '( t) ( st, ) ds t ϕ b Proof : i) Let t [ cd, ] d > Sice ϕ is cotiuous o [ b, ] [ cd, ], we hve ϕ is uiforly cotiuous o [ b, ] [ cd, ] Therefore, there is δ > such tht for ech s [ b, ], we hve, 9

36 ϕ( st, + h) ϕ( st, ) < Now, for h < δ we hve b b, wheever t+ h t < δ g( t + h) g( t ) ϕ( st, + h) ds ϕ( st, ) ds b [ ϕ(, + ) ϕ(, ) ] b st h st ds b ϕ( st, + h) ϕ( st, ) ds b ds b b b Thus, g( t+ h) g( t), wheever h < δ ie g is cotiuous t t [ cd, ] Sice t is rbitrry eleet of [ cd, ], we hve g is cotiuous o [ cd, ] ii) Suppose tht Let t [ cd, ] d > Deote, ( st, ) ϕ ( st, ) ϕ t ϕ exists d cotiuous t Agi sice ϕ ( st, ) is uiforly cotiuous o [ b, ] [ cd, ], δ > such tht for ech s [ b, ], we hve ϕ( st, ) ϕ( st, ) <, wheever t t < δ Thus, for t t < δ d s [ b, ] ie [ ϕ (, τ ) ϕ (, ) ] t, we hve [ (, ) (, ) ] ϕ sτ ϕ st dτ < dτ t t t s st dτ < t t, wheever t t t < δ d s [ b, ] t () LetΦ ( t) ϕ( st, ) tϕ( st, ), for soe fixed s [ b, ] The Φ '( t) ϕ( st, ) ϕ( st, ) ie Φ ( t) is priitive of ϕ( st, ) ϕ( st, ) 3

37 The by Fudetl Theore of Clculus for Lie Itegrls, we hve t [ ϕ τ ϕ ] t ( s, ) ( st, ) dτ Φ( t) Φ( t ) Iequlity () becoes, [ st st ] ( t t ) ϕ(, ) ϕ(, ) ϕ ( st, ) [ ] ϕ( st, ) ϕ( st, ) t t ϕ ( st, ) < t t () Now, b b b b g( t) g( t) ϕ( st, ) ds ϕ( st, ) ds ϕ( st, ) ds ϕ( st, ) ds t t t t ϕ( st, ) ϕ( st, ) t t ϕ ( st, ) ds b t t b ϕ( st, ) ϕ( st, ) t t ϕ ( st, ) t t ds Thus, for ech >, δ > such tht g( t) g( t ) t t b < ds ( b ) (By equtio ()) b ϕ( st, ) ds < ( b ), wheever ie g is differetible t t d hece o [c, d] b Next, g '( t) ϕ( st, ) ds ϕ( st, ) ds t As ( st, ) ϕ ϕ is cotiuous, we hve g ' is cotiuous t Hece, gis cotiuously differetible b < t t < δ 3

38 Exple 4 Prove tht π is e ds π, whe < is e Solutio: Defie g :[,] by By Leibit rule, g( t) π is e ds, t [,] is e t π is e g '( t) ds is t e t π e is is ( e t) ds i Let Φ ( s) is e t The Φ '( s) e is is ( e t) Thus, g '( t) Φ( π ) Φ () (By fudetl theore of clculus for lie itegrls) i i π i i e t e t ie g '( t), t [,] Therefore, g is costt fuctio I prticulr, we hve g() g() ie π is π e ds ds e is π is e ie ds π is e Exercise 5 Show tht d πi, whe < it (Hit: put e d use bove exple) The followig theore is ow s Cuchy Itegrl Forul 3

39 it Theore 6 Let f : G be lytic d suppose B(, r) G( r> ) If ( t) + re, t π, the f ( w) f dw πi, for < r w Proof Without loss of geerlity ssue d r Tht is we y ssue tht B(,) G Let B(,) ; we hve to show tht f ( w) f dw πi w π is is f ( e ) e i is ds πi e ie π f π f ( e ) e is e is is ds ie π f ( e ) e is e is is f ds is is f ( + t( e )) e Let ϕ( st, ) f is e where s π d t is is is Sice + t( e ) ( t) + te ( t) + t e <, ϕ is well defied d cotiuously differetible Now defie g :[,] by π g( t) ϕ( st, ) ds The by Leibit rule g h s cotiuous derivtive π g '( t) ϕ( st, ) ds t 33

40 π is is f ( + t( e )) e is t e f ds π '( ( is is f + t e )) e ds Let is is is Φ ( s) it f ( + t( e )) the '( s) f '( t( e )) e Φ + for < t g '( t) Φ( π ) Φ () (By fudetl theore of clculus for lie itegrls) πi i it f ( + t( e )) + it f ( + t( e )) g '( t ) for < t Sice gis cotiuous o [,] we hve g '() g '( t ) for t g is costt o [,] g() g() π is f ( + e ) e is e is f ds π f e is e is f ds π is π e f ds ds is e [ ] f π π Le 7 Let be rectifible curve i d suppose tht fuctios o{ } If F u o { } lif the F li F F d F re cotiuous Proof Sice F u lif, for give > there is iteger > such tht F ( w) F( w) V ( ) < for ll w { } d 34

41 Therefore F F ( F F) whe F( w) F ( w) dw The followig theore gives the Tylor s series expsio of lytic fuctio: Theore 8 Let f be lytic i B(, R ) The where! f for < R f d this series hs rdius of covergece R Proof Sice f is lytic i B(, R ), the there is < r< R such tht B(, r) B(, R) it Let ( t) + re, t π the by Cuchy itegrl forul, we hve f ( w) f dw πi for < r w () Now w ( w ) ( ) w w w w w w w w () ( w ) 35

42 Sice { } is copct d f is cotiuous o { }, f bouded o { } { } Let M sup f ( w) : w { } The f ( w)( ) f ( w) ( ) M ( ) ( w ) ( w ) + + ( w ) ( w ) M ( ) r r Let M M ( ) r r, the M < s < r f ( w)( ) By Weierstrss M- test the series coverges uiforly to + ( w ) w { } I view of (), () d Le f ( w) f ( w)( ) f dw dw πi w i ( w ) + π f ( w) w for f ( w)( ) + πi ( w ) dw f ( w) dw ( ) + πi ( w ) f ( w) where dw πi + ( w ), < r Thus f hs power series expsio i B(, R )! f, so tht vlue of is idepedet of, hece idepedet of r 36

43 Moreover, s < r< R is rbitrry, we hve f ( ) clerly rdius of covergece for < R R Corollry 9 If f : G is lytic d G the where R d(, G) Proof Sice f for < R { } { } R d, G if d, G : G if : G, B(, R) G Sice f is lytic i G, f is lytic o B(, R ) Hece by Tylor s theore, f for < R Corollry If f : G is lytic the f is ifiitely differetible Proof Suppose f : G is lytic, the for y G, f hs Tylor s series expsio bout f for R The by theore, f is ifiitely differetible <, R d(, G) Corollry If f : G is lytic d B(, r) G the f! f ( w) πi ( w ) + dw it where ( t) + re, t π Proof Suppose f : G is lytic d B(, r) G it Let ( t) + re, t π The by Tylor s theore, f ( w) f where dw πi + ( w ) 37

44 We lso hve, f!! f ( w) πi ( w ) f dw + i e Exple Evlute the itegrl d it where ( t) re, t π i Solutio: Let f e, the f is lytic fuctio We hve f ()! f () πi ( ) + d ie i! e πi i d i e d π si Exple 8 Evlute the itegrl 3 d it where ( t) re, t π Solutio: Let f si, the f is lytic fuctio We hve f ()! f () πi ( ) + d si si() 3 d πi si Therefore 3 d 38

45 Exple 3 Evlute the itegrl d it where ( t) + re, t π Solutio: Let f ( ), the f is lytic fuctio We hve () f! f πi ( ) + d πi d Therefore, d πi Exple 4 Evlute the itegrl e + si d it where ( t) re, t π Solutio: Let f e + si, the f is lytic fuctio We hve f ()! f () πi ( ) + d! e + si e + si d πi Therefore e + si d π i The followig theore is ow s Cuchy s Estite Theore 5 Let f be lytic i B(, R) d suppose tht f M for llf i B(, R ) The M R! f Proof Sice f is lytic i B(, R ), the there is < r< R such tht B(, r) B(, R) The by corollry, f! f ( w) πi ( w ) it dw where ( t) + re, + 39 t π

46 Therefore f! f ( w) πi ( w ) + dw! f ( w) + π w dw! f! M dw π + r! M πr + π r M r Sice < r< R is orbitrry, lettig r R we get M R! f Cuchy s estite leds us to Cuchy s Theore: Theore 6 Let fbe lytic i the dis B(, R ) d suppose tht is closed rectifible curve i B(, R ) The f Proof Sice f is lytic i B(, R ), it hs power series expsio Let f for < R F () () li + Sice li Sup li Sup li li Sup + + Thus series () d () hve se rdius of covergece Therefore,F is defied d lytic i B(, R ) 4

47 Hece F ' + ( ) ( ) f + ie F is priitive of f If :[ b, ] the f F ' F( ( b)) F( ), Sice is closed curve ( b) EXERCISES ) Evlute the itegrl d it, where ( t) e, t π, is soe positive iteger ) Evlute the itegrl d,, where is closed polygol curve [-i, +i, -+i, --i, -i] 3) Let, δ be polygos [, +i, i] d [, i] respectively Evlute the itegrl over s well s δ i e 4) Evlute d it, where ( t) e, t π si 5) Evlute 3 d it, where ( t) e, t π d 6) Let G be coected set d f : G be lytic fuctio If f() is rel for ll i G, the prove tht f is costt fuctio 7) Prove bove exercise for ) f() is igiry uber for ll d b) f() with costt odulus 4

48 UNIT - IV FUNDAMENTAL THEOREM OF ALGEBRA AND MAXIMUM MODULUS THEOREM I this uit we prove Liouville s theore use it to prove fudetl theore of lgebr We lso prove xiu odulus theore Defiitio A etire (itegrl) fuctio is fuctio which is defied d lytic i the whole coplex ple Note f e,si,cos re etire fuctios All polyoils re etire fuctios Theore If f is etire fuctio the f hs power series expsio ifiite rdius of covergece Proof For y R>, B(, R) The f is lytic i B(, R ) f with By Tylor s theore, f, for R < Sice R> is rbitrry, rdius of covergece is ifiite Followig theore is ow s Liouville s theore Theore 3 If fis bouded d etire fuctio the f is costt Proof Sice f is bouded d etire fuctio, f M d for, R>,f is lytic i B(, R ) By Cuchy s Estite, we hve f M R! 4

49 I prticulr for we hve f M R '! Sice R is rbitrry, s R, we get f ' Therefore, f ' ( ) for y Thus f is costt Thus we c prove the Fudetl theore of Algebr: Theore 3 If p( ) is o costt polyoil the there is coplex uber with p( ) Proof Let p( ) is o costt polyoil d tht p for y Let f p the f is etire fuctio ( p ( ) is etire d p ) Sice p( ) is o costt polyoil, ssue tht ( ) li p li p li f li p Therefore, for there is R> such tht f <, wheever > R Tht is f ( ) <, wheever > R Sice f is cotiuous o closed bouded dis B(, R), f is bouded o B(, R ) Therefore, there is M > such tht f M for B(, R) Tht is f M, wheever R Thus f x{, M}, for ll This f is bouded etire fuctio Hece by Liouville s theore f is costt d cosequetly p is costt Which cotrdicts our ssuptio Hece the theore 43

50 Defiitio 4 Let f : G be lytic d G stisfies f ( ) the is ero of f of ultiplicity if there is lytic fuctio g : G such tht f ( ) g, where g Corollry 5 If p( ) is polyoil d,,, re its eros with j hvig ultiplicity j the p c( ) ( ) for soe costt c d is the degree of p Proof Sice,,, re eros of p hvig ultiplicities,,, respectively, there exists polyoil g( ) such tht p ( ) ( ) g, where g( ), ( j ) j Therefore by fudetl theore of lgebr, g( ) is forced to be costt Let g c for soe c p c ( ) Obviously, degree of p( ) is Theore 6 Let G be coected ope set d let f : G be lytic fuctio The the followig re equivlet stteets: () f ; (b) { G : f } hs liit poit i G ; (c) there is poit i G such tht f ( ) for ech Proof ( b) suppose f, the { G : f } G, which is ope Hece, every poit of G is liit poit of G Thus { G : f } ( b) ( c) hs liit poit i G 44

51 Suppose tht, A { G : f } hs liit poit i G, the there is sequece { } of poits i A such tht li Sice f is cotiuous, f li f ( ) ( A f ( ) ) Now suppose tht, there is iteger such tht f f f ( ) ' d ( f ) Sice f is lytic i G, there is By Tylor s theore B R G R> such tht f is lytic i ( ; ) f for < R where! ( ) f Sice ( ) f f ' f d ( f ) we hve f ( ) ( ) ( ), the g is lytic i ( ; ) Let g f ( ) g d g Sice g is cotiuous i ( ; ) As is liit poit of A, B R d B R, there is R> r> such tht B r ; A { } φ Let b B( r ; ) A { }, the b B( r ; ) Therefore, b B( r) d b A { } A B r g i ( ; ) ; g( b) d b A { } A f ( b) g( b) which gives the cotrdictio to our ssuptio Hece o such iteger c be foud Thus f ( ) for ech ( b) ( c) Suppose there is i G such tht f ( ) for ech Let H { G : f } the H φ 45

52 Now we cli tht H is both ope d closed : Let H, the there is B( ; R) G By Tylor s theore f for < R R> such tht ( ; ) B R GR> The f is lytic i where! f Sice f ( ) for ech, ech f ( ) i B( ; R ) ie f ( ) i B( ; R ) B( ; R) H f i ( ; ) Thus for H, the there is Hece H is ope B R B R H R> such tht ( ; ) Now let be liit poit of H, the there is sequece { } i H such tht li Sice f is cotiuous, li f f Therefore ( f ) ( ) iplies H Thus H H Hece H is closed Thus H is ope s well s closed subset of coected set G Hece by property of coectedess H G Therefore, ( f ) G, Tht is f G Hece f o G Corollry 7 If f d g re lytic o regio G, the f g hs liit poit i G Proof Let h f g G, which is lytic i G iff { G : f g } 46

53 The h o G { G : h } ie f g ie f g ie f g hs liit poit i G o G { G : f g } o G { G : f g } o G { G : f g } hs liit poit i G hs liit poit i G hs liit poit i G Corollry 8 If f is lytic o ope coected set G d f is ot ideticlly ero the for ech i G with f ( ) there is iteger d lytic fuctio g : G such tht g d f ( ) g for ll i G Tht is ech ero of f hs fiite ultiplicity Proof Let f be lytic o ope coected set G Sice f d f ( ) for soe i G, there is positive iteger such tht f f ' f d ( f ) Now we defie g : G, by f g ( ) for ( f ) for ()! Therefore g is lytic o G { } Now to show tht g is lytic o G it eed oly to show g is lytic i eighborhood of Sice f is lytic i G, there is B R G R> such tht f is lytic i ( ; ) By Tylor s theore f for < R where! ( ) f Sice ' f f f d ( f ) we hve f ( ) ( ) ( ) 47

54 , the h is lytic i ( ; ) Let h f ( ) h d Thus fro () g h i B( ; R ) Therefore g is lytic i B( ; R ) ( f ) h! B R d Hece g is lytic i G with f ( ) g d g h Corollry 9 If f : G is lytic d ot costt, G d f ( ) the there is R> such tht B( ; R) G d f for < < R Tht is eros of re isolted Proof Let f : G be o-costt lytic fuctio with f ( ) for soe G The by corollry there is lytic fuctio g : G d iteger such tht f ( ) g d g Sice g is lytic, g is cotiuous o G Therefore, there is R> such tht g i ( ; ) Hece f ( ) g for < < R B R G ie for < R We ow prove Mxiu Modulus Theore: Theore Let G is regio d f : G is lytic fuctio such tht there is poit i G with f f for ll i G, the f is costt Proof Sice f : G is lytic fuctio, there is The by Cuchy Itegrl forul r> such tht ( ; ) B r G f ( w) it f dw πi for < r d ( t) + re, t π w π it f ( + re ) it rie dt it πi + re π it f ( + re ) dt π 48

55 π it f f ( + re ) dt π π f dt π s f f for ll i G f Therefore π it f f ( + re ) dt f π π it f f ( + re ) dt π π it f f ( + re ) dt π () it Sice f f ( + re ) for ll t it Therefore fro () we hve f f ( + re ) for ll t it Let f α, the f ( + re ) α for ll t Sice r> is rbitrry, we hve f α B r for ll i ( ; ) Tht is f ps whole dis B( r ; ) G ito the circle α where f α Therefore f hs costt odulus o B( r ; ) d hece fis costt o ( ; ) Let f c o B( r ; ) G the { G : f c} B( r ; ) Thus f c o G, ie f is costt o G B r hs liit poit i G Theore If :[,] is closed rectifible curve { } d πi is iteger Proof Defie g :[,] by g( t) t '( s) ds ( s), the 49

56 where is closed rectifible curve so tht g is well defied Therefore g () d '( s) g() ds d ( s) Also '( t) g '( t) ( t) for t The d e g ( t ) t e g ( t ) t e g ( t ) g t t ( ) ' ' ( ) dt g ( t) '( t) e '( t) ( ( t) ) ( t) Therefore Hece e g t ( ( t) ) is costt fuctio g () g () e e ( () ) ( () ) g () e e ( () () ) πi g () e e ( is iteger ) Therefore g() πi Therefore, π i d Hece d πi Defiitio If is closed rectifible curve i the for { } The the iteger d πi is clled the idex of with respect to the poit Defiitio 3 A subset D of etric spce X is clled copoet of X, if it is xil coected subset of X 5

57 Theore 4 Let be closed rectifible curve i The is costt for belogig to copoet of G { } ) ( ; ) b) ( ; ) for belogig to the ubouded copoet of G Proof Defie f : G by f d (, ) πi i G Cli: f is cotiuous Let G d r d { }, > For y > we choose δ > such tht Therefore b < δ < r f ( b) f d πi b π b ( b)( ) d b π d ( b) ( ) Now r> r d b ( ) ( b ) b > r r r Thus f ( b ) f ( δ ) d δ < π V r r πr Therefore, for y > there is wheever b < δ Thus f is cotiuous i r πr < δ <,, such tht f ( b) f V ( ) 5 <,

58 ) Let D be copoet of G the D is ope d coected Sice f is cotiuous, f ( D ) is coected Sice f is iteger vlued d subset of set of ll itegers which re coected re precisely sigleto sets Therefore f ( D) { } Tht is f i D for soe iteger Hece ( ; ) is costt for belogig to D b) Let U be ubouded copoet of G { } { : } G > R U, the there is R> such tht For > choose such tht > R d V ( ) > π for ll o { } the ( ; ) d πi π d d π πv ( ) Therefore ( ; ) < Sice ( ; ) is iteger ( ; ) < ( ; ) for soe i U Sice f ( ; ) is costt o U, we ust hve ( ; ) for ll i U 5

59 EXERCISES ) Let f : C C be etire fuctio Suppose for soe R >, > R iplies f() M, for soe costt M The prove tht f is polyoil of degree t ost ) Let f : G C be lytic fuctio defied o regio G with f() f(), for ll i G Show tht either f or f is costt fuctio 3) Let f d g be lytic fuctios defied o the regio G If fg o G, prove tht either f or g 4) Show by exple tht ( ; ) for closed rectifible curve i C, where { } 53

60 UNIT - V WINDING NUMBERS AND CAUCHYS INTEGRALTHEOREM I the lst uit we prove tht d πi is iteger We shll deote this iteger by ( ; ) d clled it is widig uber or Idex of closed curve roud I this uit we discuss Cuchy s itegrl forule Le Let be rectifible curve d suppose φ is fuctio defied d cotiuous o { } For ech let { } d ' + F F F φ( w) dw for { } ( w ) The ech F is lytic o Proof Let F rectifible curve First we cli tht ech φ( w) dw for { } ( w ) F is cotiuous: Let G { } d r d { }, > For y > we choose δ > such tht Therefore φ( w) φ( w) F F ( w ) ( w ) where φ is cotiuous fuctio d is dw < δ < r φ( w) dw () ( w ) ( w ) { } Sice φ is cotiuous fuctio o copct set { }, we hve M sup φ( w) : w { } 54

61 Ad ( w ) ( w ) ( w ) ( w ) ( w ) ( w ) ( ) ( w ) ( w ) + Also w r> r d w ( w ) ( ) w > r r r Therefore ( w ) ( w ) ( w ) ( w ) + ( ) ( w ) ( w ) + Thus () gives < δ δ + r ( r ) ( r ) + ( ) ( ) + + F F < Mδ dw Mδ V ( ) r r Therefore, for y > there is ( r ) + δ i r, < <, such tht F F MV ( ) <, wheever < δ Thus F is cotiuous o G { } for y Now to show Cosider F F : ' + φ( w) φ( w) F F ( w ) ( w ) dw 55

62 ( ) + φ( w) ( w ) ( w ) + dw Thus ( ) + + φ( w) w F F ( ) ( w ) dw + Sice { } d φ( w) ( w ) cotiuous o { } ( ) + φ( w) w dw is cotiuous ( w ) for ech, ech itegrl Hece lettig we get ( ) + F F φ li ( ) w w li ( w ) dw ( ) ( w ) ( ) + + φ( w) w φ( w) w li dw dw ( w ) F F w w li φ φ dw + + ( ) w w dw Therefore, F F ' for ll { } + Thus, F is differetible for y Sice F + is cotiuous, F ' is cotiuous Therefore F is cotiuously differetible Hece F is lytic o { } 56

63 We ow prove Cuchy s Itegrl forule: Theore ( First Versio) Let G be ope subset of the ple d f : G be lytic fuctio If is closed rectifible curve i G such tht ( ; w) for ll w i G, the for i G { } f ( ; ) f d πi Proof Defie φ :G G by φ(, w) The φ is cotiuous f ( w) f w if w f ' if w Let H { w : ( ; w) } ope Moreover Now, defie g : by g φ(, w) dw Sice ( ; w) is cotiuous iteger vlued fuctio, H is G H Thus G H if G f ( w) dw if H w If G H the φ(, w) dw f ( w) f dw w f ( w) dw f ( ) w w dw f ( w) dw f ( ) ( ; ) π i w f ( w) dw f ( ) π i w f ( w) dw w Therefore g is well defied fuctio Thus by le g is lytic o, hece gis etire fuctio 57

64 Sice H cotis eighborhood of ifiity, we hve Sice { } is copct, f is bouded o { } Hece there is M > such tht f ( w) M for ll w { } li w uiforly for w { } Therefore, f ( w) f ( w) dw w w dw M w dw Hece f ( w) li g li dw w Therefore, there is R> such tht g for > R Sice, g cotiuous o copct set B(, R ), g is bouded o B(, R ) Thus g is bouded etire fuctio Hece by Liouville s Theore g is costt Sice li g we ust hve g Thus f ( w) f dw w for ll i G { } f ( w ) dw f ( ) dw w w for ll i G { } f ( w) dw f ( ) π i ( ; ) w for ll i G { } Thus f ( ; ) f d πi for ll i G { } 58

65 Theore 3 (Secod Versio) Let G be ope subset of the ple d f : G be lytic fuctio If,,, re closed rectifible curves i G such tht ( ; w) + ( ; w) + + ( ; w) for ll w i G, the for i G { } f ( ; ) f d πi Proof Defie φ :G G by φ(, w) The φ is cotiuous f ( w) f w if w f ' if w Let H { w : ( ; w) + ( ; w) + + ( ; w) } Sice ( ; w) is cotiuous iteger vlued fuctio, H is ope Moreover Now, defie g : by G H Thus G H if G g φ(, w) dw f ( w) dw w if H If G H the φ(, w) dw f ( w) f dw w f ( w) dw f ( ) w w dw f ( w) dw f ( ) ( ; ) π i w 59

66 f ( w) f ( w) dw f πi ( ; ) dw f πi w w f ( w) dw w Therefore g is well defied fuctio Thus by le g is lytic o, hece gis etire fuctio Sice H cotis eighborhood of ifiity, we hve Sice ech { } is copct, f is bouded o { } Hece li g li w uiforly for w { } Therefore, there is R> such tht g for > R Sice g cotiuous o copct set B(, R ), g is bouded o B(, R ) Thus g is bouded etire fuctio Hece by Liouville s Theore g is costt Sice li g we ust hve g Thus f ( w) f dw w for ll i G { } f ( w) dw f ( ) dw w w for ll i G { } f ( w) dw f ( ) π i ( ; ) w for ll i G { } Thus f ( ; ) f d πi for ll i G { } 6

67 Theore 4 Let G be ope subset of the ple d f : G be lytic fuctio If,,, re closed rectifible curves i G such tht ( ; w) + ( ; w) + + ( ; w) for ll w i G, the f Proof Let { } d F ( ) f The F is lytic ig Hece by Cuchy s itegrl theore, F ( ; ) F d πi πi ( ; ) ( ) f d f d Thus f Theore 5 Let G be ope subset of the ple d f : G be lytic fuctio If,,, re closed rectifible curves i G such tht ( ; w) + ( ; w) + + ( ; w) for ll w i G, the for i G { } f ( ; )! f d πi Proof By Cuchy Itegrl forul we hve f ( ; ) f d πi for ll i G { } 6

68 Hece f ( ; ) d f d d πi f d πi Therefore, f ( ; )! f πi + ( ) f d πi d Corollry 6 Let G be ope subset of the ple d f : G be lytic fuctio If is closed rectifible curve i G such tht ( ; w) for ll G { }! f ( ; ) πi f d + ( ) w i G, the for i Defiitio 7 A closed polygol pth hvig three sides is clled trigulr pth Theore 8 Morer s Theore Let G be regio d let f : G be cotiuous fuctio such tht f for every trigulr pth T i G ; the f is lytic i G T Proof To prove tht f is lytic i G we hve to prove tht f is lytic o ech ope dis cotied i G Hece without loss of geerlity we y ssue tht G B( ; R) Now defie F : G by where [, ] F f ( wdw ) [ ] Fix B( ; R) is the lie seget joiig to, the for y i G, T [,,, ] be trigulr pth ig 6

69 Therefore by hypothesis f T f + f + f [, ] [, ] [, ] f f + f [, ] [, ] [, ] F f ( wdw ) + F( ) [, ] Thus F F( ) f ( ) f ( wdw ) f ( ) [, ] [ f ( w) f ] dw [, ] Sice f is cotiuous i G, for y > there is δ >, such tht f ( w) f ( ) < wheever w < δ Therefore F F( ) f ( ) f ( w) f ( ) dw < dw [, ] [, ] Thus F F( ) li f F '( ) f ( ) Sice is rbitrry, we hve Sice f is cotiuous, F ' f i G F ' is cotiuous o G Therefore F is cotiuously differetible, tht isf is lytic Hece F ' f is lso lytic i G 63

70 Sigulrities Defiitio 9 A fuctio f hs sigulrity t if f is ot lytic t Ex Ex si,, e cos ( ) hs sigulrity t hs sigulrity t the poits,, ±, ±, (+ ) π Defiitio A fuctio f hs isolted sigulrity t if there is R> such tht f is lytic i B( ; R) { },otherwise is o-isolted sigulrity of f Ex Ex si ( π ) ( )( ), re isolted sigulrities d is o-isolted sigulrity,, re isolted sigulrities There re three ids of isolted sigulrities A) Reovble sigulrity B) Pole C) Essetil Sigulrity Defiitio A isolted sigulrity t of fuctio f is reovble sigulrity if there is < < R Ex R > d lytic fuctio : (, ) si f hs reovble sigulrity t Ex f e hs reovble sigulrity t g B R such tht g f i Theore If f hs isolted sigulrity t, the the poit is reovble sigulrity iff li( ) f ( ) 64

71 Proof Suppose is reovble sigulrity, the there is R> d lytic fuctio : (, ) g B R such tht g f i < < R Therefore, li( ) f li( ) g g ( siceg is cotiuous ) Coversely suppose tht li( ) f ( ) Sice f hs isolted sigulrity t, there is Defie R> such tht f is lytic i B( ; R) { } ( ) f if h if Clerly h is lytic i B( ; R) { } d cotiuous t Now to provef hs reovble sigulrity t we hve to prove tht h is lytic i B( ; R ) Cli: h is lytic i B( ; R ) To prove this we use Morers theore Let T be the trigle i B( ; R ) d deote iside of T log with T Cse : Whe Sice h is lytic i B( ; R) { } Cse : Whe is vertex of T d T T, by Cuchy theore h Let T [ bc,,, ] be trigle with s oe of the vertex For x [ b, ] d y [ c, ] let P [ xbc,,, y, x], the P d by Cuchy theore h Let T [, x, y, ] the h h+ h h T T P T P T Sice h is cotiuous d h, for y > there is δ > such tht < δ h < lt for Now we choose x, y such tht x, y B ( ; δ ) Therefore h d h d h d < lt ( ) lt ( ) T T T 65

72 Hece h T Cse 3 Whe lies o or iside T I this cse we c costruct trigle s show with belogig to vertex of ech costructed trigle Therefore usig cse we ust hve h for y trigulr pth T i ( ; ) Thus h T i B( ; R ) T B R Hece by Morers theore h is lytic y T x T T Cse : lies outside T Cse : is vertex of T Cse 3 : lies iside T Corollry 3 A isolted sigulrity of fuctio f t is reovble sigulrity iff f is bouded i the eighborhood of Proof Let is reovble sigulrity of f the there is R> d lytic fuctio : ( ; ) g B R such tht g f i < < R Sice g is cotiuous t, g( ) is fiite Hece g is bouded i eighborhood of Therefore f is bouded i the eighborhood of Coversely, suppose f is bouded i the eighborhood of, the there is M > such tht f M i < < δ Therefore ( ) f M s Thus li( ) f ( ) Hece f hs reovble sigulrity t 66

73 Corollry 4 A isolted sigulrity of fuctio f t is reovble sigulrity iff li f ( ) c Proof Let is reovble sigulrity t the there is R> d lytic fuctio : ( ; ) g B R such tht g f i < < R Therefore li f li g g c (sy) Coversely, suppose li f ( ) c the li( ) f li( )li f c Thus li( ) f ( ) Hece f hs reovble sigulrity t Defiitio 5 A isolted sigulrity t of fuctio f is pole if li f ( ) Ex f hs pole t Ex f ( )( i) hs pole t, i Theore 6 If G is regio with i G d if f is lytic o G { } with pole t, the there is positive iteger d lytic fuctio g : G such tht g f ( ) Proof Suppose is pole of f the li f ( ) Therefore li f The f hs reovble sigulrity t The there is lytic fuctio h : B( ; R) such tht h whe < < R f Now we defie if h f if 67