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Homework 1 collaboration area

Feel free to toss around ideas here.--Steve Bell

Here is my favorite formula:

$ f(a)=\frac{1}{2\pi i}\int_\gamma \frac{f(z)}{z-a}\ dz. $

Problem 1

Problem 2

Problem 3

Difference quotient should include a special case when $ f(z)=f(z_0) $.

Problem 4

Problem 5

Use Problem 4.

Problem 6

Just throwing some stuff here for test purpose:

About the trick in the Problem 6, one direction is easy;

The other direction can be proved using a trick by considering $ r-\epsilon $ where $ \epsilon>0 $ is some arbitrarily small quantity. This yields a convergent geometric series, which serves as an upper-bound of the original absolute series. Finally, let $ \epsilon $ go to zero. Result from Problem 5 is involved.

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