From: Robin Chapman <rjc@maths.ex.ac.uk>
Newsgroups: sci.math
Subject: Re: Analytic functions with growth rate < r
Date: Thu, 12 Mar 1998 02:21:00 -0600

In article <6e6962$gpa$1@nnrp1.dejanews.com>,
  bobs@rsa.com wrote:
>
> In article <6e5umc$9nd$1@nnrp1.dejanews.com>,
>   I wrote:
> >
> > One could give an alternative proof of this by invoking the
> > Casaroti-Weierstrass theorem, but this would be cracking a very small
> > nut with a sledgehammer.
> >
>
> Please educate me. What is this theorem? I may know its statement, but
> I certainly don't know or remember it by name.
>

This states that in a neighbourhood of an isolated essential singularity
the set of values taken by an analytic function is dense in C.
Applying this to z^{-a}f(z) for an integer a where f has an essential
singularity at infinity, it shows that f(z) isn't O(|z|^a). Of course
it's implied by the great Picard theorem, but that is much deeper.

Robin Chapman

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