# Cauchy estimates

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:

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