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Unformatted text preview: C is a closed contour (not necessarily simple)) lying inside a simply connected domain D, and f is analytic on D, then Û Ù O f(z) dz = 0 ı C (We show this for the case of finitely many self-intersection points). 15 4. If f is analytic throughout a simply connected domain D, then f has an antiderivative in D. (We construct the antiderivative by brute force.) Examples Ûz Ù (A) O e dz = 0 for any old closed curve C. ı C Û dz Ù (B) O 2 = 0 for any closed curve C not including 0. ız C Û dz Ù (C) O ı z = 2πi for every simple contour enclosing 0. (Consequence 2) C Û dz Ù (D) O ı z-Ω = 2πi for any simple closed contour about Ω . We can evaluate this using C it Consequence 2 and taking C to be the circle Ω + e . Û dz Ù (E) O ı (z-Ω)n = 0 if n is any integer other than 1. (Evaluate it directly for a circle). C Û 5z + 7 Ù (F) Evaluate O 2 ı z + 2z - 3 dz where C is the circle |z-2| = 2 (Use partial fractions) C In general, we have Consequence 5. if f is not defined at z1, ..., zk, and C is a simple contour surrounding them all, then Û Û Û Ù Ù Ù O f(z) dz = O f(z) dz + ... + O f(z) dz ı ı ı C C C 1 k where the Ci are simple contours around the zi. Û Ù Example Apply th...
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This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.

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