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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

On Fri, 08 Mar 2013 13:00:32 -0800, W^3 <82ndAve@comcast.net> 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...

Date Subject Author
3/5/13 Paul
3/5/13 bacle
3/5/13 W^3
3/5/13 Scott Berg
3/5/13 J. Antonio Perez M.
3/6/13 Scott Berg
3/6/13 Frederick Williams
3/6/13 Robin Chapman
3/6/13 David C. Ullrich
3/6/13 Paul
3/7/13 AP
3/7/13 David C. Ullrich
3/7/13 William Hughes
3/8/13 quasi
3/8/13 William Hughes
3/8/13 quasi
3/8/13 AP
3/8/13 David C. Ullrich
3/8/13 W^3
3/9/13 David C. Ullrich