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. $

Here is Stirling's Formula:

$ \lim_{n\to\infty}\frac{n!}{\sqrt{n}\left(\frac{n}{e}\right)^n} =\sqrt{2\pi}. $--Steve Bell 10:28, 19 January 2011 (UTC)

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.

Back to the MA 530 Rhea start page

To Rhea Course List

Alumni Liaison

Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett