
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 >CasatoriWeierstrass, 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 CW 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, CW is another thing that's sort of in the direction of Big Picard but much more elementary; CW is what was actually lurking in my head somewhere.
Aargh. Time to clean out my desk...

