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.