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Complex Variables

by: Miss Cloyd Cronin

Complex Variables MAT 3223

Miss Cloyd Cronin
GPA 3.84

Dmitry Gokhman

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Dmitry Gokhman
Class Notes
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This 15 page Class Notes was uploaded by Miss Cloyd Cronin on Thursday October 29, 2015. The Class Notes belongs to MAT 3223 at University of Texas at San Antonio taught by Dmitry Gokhman in Fall. Since its upload, it has received 18 views. For similar materials see /class/231335/mat-3223-university-of-texas-at-san-antonio in Mathematics (M) at University of Texas at San Antonio.

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Date Created: 10/29/15
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4 had n 12M i 3 quot 1 7 1 m zquotquot i w 39539ch 1ng Q 751Oi 10440398 311 K 551 M M i l I T i u m 575 3 l 5751 39llu 9 Figum 422 5 Evaluate fW d along the path indicated in Figure 422 Ml39 I hJ L L MALHL um FEAR F dwzq39ih Lira SJ 110 1T l e L 2 1 am E 39R39C llb39 39l lug MT Cmvh NHL aquot inv db kc MAL quotKn ulnar 39 39 39 139 m n a M m by Lua 1ru ltr 4 1 DM39IT i 1 Ht V ma13913 WU u 9 mm n n m r anqr In A3 llt mina avari I 5 Jen 3531 n Pan ff 1 mg rn a u NHAnnWlud J e n H MUN um lePAInrJLul jl If Vlad 13 m 5 Timid ful l 35 N4quot bx 195 n pawl 31 a 5 M Na g H 8 9a a Ea em a as E V n E A my S 87 as as a a E t n E A 3 n r WTVN Fin W Wu HS y n 131 vule hi Amaiy m n wn w u zmw 39 iii m 3915 A511 H313 r9133 Nairnx P PFUTL r 33 quotKay L h FTP tp 03L JEnFA n fA 83 5 n u 23 Ti gnu u Lur tu h mduw rt 3 W nquot was 1 auLAN o n NC Chapter Five Cauchy s Theorem e y t hh hn D We eay Ch he hnmntnphe to C2 hn D 1fthere he a eontanhhohhe function H s D Where she the square S 15 0 5 515 1 ehheh that 1100 deeenbee C and 1101 deeenbee C2 and oh eaeh xed e the funetaon H e eeenbee a e1oeeo1 euwe C hn D 2 hn 2 he homotophe to C hn D Juet observe that the function Kte e Ht 1 e 5 he ahomotopy 1t he eonvenhent to eonehdeh a pohnt to be a e1oeeo1 euwe The pohnt e he a deeenbed by a eonetant funetaon yt e We thhhe speak of a e1oeeo1 euwe C behng homotophe to a constantiwe eometamee eay Che ennthaetjhleto a pohnt Emotaonany the fact that two e1oeeo1 euwee are homotophe hn D meane that one ean be eontanhhohhe1y deformedmw the otheh hn D Ha 13 Hg 0 Example Let D be the annhh1ah heghon D e z 1 lt 121lt 5 Suppoee Ch he the ehhe1e deeenbedby mt 217quot 0 s t s 1 and C2 he the ehhe1e deeenbed byy2t 421711 0 s t 1 Then Hts 2 2equot he a homotopy hn D between Ch and C2 Suppose C3 he the eame ehhe1e ae C2 but whth the oppoehte onentataon that he a deeenpthon he ghven by V30 e 424quot 0 s t s 1 A homotopy between C and C3 he not too eaey to n fact ht he not poeehb1e1 The moha1 onentataon eohhnte From now on the tehm quote1oeeo1 euwequot wh11 mean an onented e1oeeo1 euwe Another Example Let D be the set obtained by removing the point z 0 from the plane Take a look at the picture Meditate on it and convince yourself that C and K are homotopic in D but 1quot and A are homotopic in D while K and lquot are not homotopic in D C A C Exercises 1 Suppose C1 is homotopic to C2 in D and C2 is homotopic to C3 in D Prove that C1 is homotopic to C3 in 2 Explain how you know that any two closed curves in the plane C are homotopic in C 3 A region D is said to be simply connected if every closed curve in D is contractible to a point in D Prove that any two closed curves in a simply connected region are homotopic in D 52 Cauchy s Theorem Suppose C1 and C2 are closed curves in a region D that are homotopic in D and suppose f is a function analytic on D Let Htv be a homotopy etween C1 and C2 For each s the function y5t describes a closed curve C in D Let 5 be given by IQ I zdz C Then 1 13 J39fHz snwdz 0 Now let s look at the derivative of s We assume everything is nice enough to allow us to differentiate under the integral 1 s h man HHB dt faxnoy 3 g l Hos 9 l t 1 l 1 6H ts 6H ts 62H ts HtsJas lJat l fHtsJMS l dt 1 0 ai Hts 6H s dt Haxn g l me g l But we know each Ht s describes a closed curve and so H0 s Hl s for all s Thus Fm Haxn g l Hmsn g l0 which means s is constant In particular 0 1 or J39 fzdz J39 fzdz c1 c2 This is a big deal We have shown that if C1 and C 2 are closed curves in a region D that are homotopic in D and f is analytic on D then I fzdz I fzdz C1 C2 An easy corollary of this result is the celebrated Cauchy s Theorem which says that if f is analytic on a simply connected region D then for any closed curve C in D 53 Mad 0 C In court testimony one is admonished to tell the truth the whole truth and nothing but the truth Well so far in this chapter we have told the truth and nothing but the truth but we have not quite told the whole truth We assumed all sorts of continuous derivatives in the preceding discussion These are not always necessaryispecifically the results can be proved true without all on I 39 quot139 I nhnm 39 39 Example Look at the picture below and convince your selfthat the path C is homotopic to the closed path consisting of the two curves C1 and C2 together with the line L We traverse the line twice once from C1 to C2 and once from C2 to C1 Observe then that an integral over this closed path is simply the sum of the integrals over C and C2 since the two integrals along L being in opposite directions would sum to zero 39Ihus iffis analytic in the region bounded by these curves the region with two holes in it then we know that jzdz jzdz Mad C c 02 Exercises 4 Prove Cauchy s Theorem 5 Let S be the square with sides x 1100 and y 1100 with the counterclockwise orientation Find


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