TEAM PROJECT. Cauchy's Integral Theorem. (a) Main Aspects. Each of the problems in

Chapter 14, Problem 14.1.53

(choose chapter or problem)

Cauchy's Integral Theorem.

(a) Main Aspects. Each of the problems in Examples 1-5 explains a basic fact in connection with Cauchy's theorem. Find five examples of your own, more complicated ones if possible. each illustrating one of those facts.

(b) Partial fractions. Write f(z) in terms of partial fractions and integrate it counterclockwise over the unit circle, where

(i) \(f(z)=\frac{2 z+3 i}{z^{2}+\frac{1}{4}}\)

(ii) \(f(z)=\frac{z+1}{z^{2}+2 z}\).

(c) Deformation of path. Review (c) and (d) of Team Project 34, Sec. 14.1. in the light of the principle of deformation of path. Then consider another family of paths with common endpoints. say, \(z(t)=t+i a\left(t-t^{2}\right)\), \(0 \leqq t \leqq 1\). and experiment with the integration of analytic and nonanalytic functions of your choice over these paths (e.g., \(z, \operatorname{Im} z, z^{2}, \operatorname{Re} z^{2}, \operatorname{Im} z^{2}\), etc).

Text Transcription:

f(z) = 2z + 3i / z^2 + 1 / 4

f(z) = z + 1 / z^2 + 2z

z(t) = t + ia (t - t^2)

0 leqq t leqq 1

z, Im z, z^2, Re z^2, Im z^2