July 23, 2012 1 Comment
We used a trick a few posts ago that I wanted to expound upon. Let , and . If we have a circle of radius centered at , since it’s compact, there is some maximum value of on . Call this . We then argued that as , because is continuous. I want to generalize this result slightly:
Theorem 1 (Cauchy Estimates) In the setup as above,
You can see that, for , the result is exactly what we already used.
Proof:Recall from the generalized Cauchy integral formula, that
Then we have the estimates: