Math144Notes

832 17 odd 17 23 29 p 837 1 5 9 11 13 15 p 842

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.

Ask a homework question - tutors are online