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Complex Variables : Rectafiable Path

Please help with the following problem.

Fix w=re^i(theta)(not equal)0 and let gamma be a rectafiable path C-{0} from 1 to w. Show there is an integer k such that
(integral)gamma z^-1 dz = log r + (theta)+ 2 pi i k

See attached file for full problem description.


Solution Preview

I assume that you have already studied or can find in your textbook or notes the basic theorems for analytic functions, in particular the Cauchy integral theorem which says that the integral around a closed path of a function which is analytic (holomorphic) everywhere inside the area bounded by the closed path is always zero.
If however you need help with these basics, you can post a request for help with them and most likely get the help.

Let us follow the Hint, and look first at the end of the procedure.

There is the curve gamma over which we are asked to calculate the integral and there is ...

Solution Summary

A rectifiable path is investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.