Contents
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
Hint: Notice that if
$ |a_n r^n|< M, $
then
$ |a_n z^n|=|a_n r^n|\left(\frac{|z|}{r}\right)^n< M\left(\frac{|z|}{r}\right)^n, $
and you can compare the series to a convergent geometric series if
$ |z|<r. $
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.
