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Topic: Elementary complex analysis
Replies: 19   Last Post: Mar 9, 2013 11:35 AM

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David C. Ullrich

Posts: 21,553
Registered: 12/6/04
Re: Elementary complex analysis
Posted: Mar 9, 2013 11:35 AM
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On Fri, 08 Mar 2013 13:00:32 -0800, W^3 <> wrote:

>Another proof: Let f(z) = sum(n=0,oo) a_nz^n. Then f(1/z) =
>sum(n=0,oo) a_n/z^n. Now f(1/z) -> oo as z -> 0. Hence by
>Casatori-Weierstrass, f(1/z) has a pole at 0. Thus f(1/z) = sum(n=0,N)
>a_n/z^n => f(z) = sum(n=0,N) a_nz^n.

Aargh. Me stupid this last week, thanks.

Another version of more or less this argument:

If f is not a polynomial then f has an essential singularity
at infinity. So C-W says the image of every nbd of
infinity is dense, in particular f does not tend to
iinfinity at infinity.

Aargh. I've been wondering why I said that
stupid thing about Little Picard. This explains that:

(i) Hmm, it's obvious from Big Picard.
(ii) But I _know_ that it's also obvious from much
less then Big Picard.
(iii) (without thinking) so it must be obvious from Little Picard.

But of course, C-W is another thing that's sort of in
the direction of Big Picard but much more elementary;
C-W is what was actually lurking in my head somewhere.

Aargh. Time to clean out my desk...

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