
Re: Elementary complex analysis
Posted:
Mar 8, 2013 10:48 AM


On Thu, 7 Mar 2013 15:43:09 0800 (PST), William Hughes <wpihughes@gmail.com> wrote:
>On Mar 6, 4:42 pm, David C. Ullrich <ullr...@math.okstate.edu> wrote: >> On Tue, 5 Mar 2013 10:29:07 0800 (PST), Paul <pepste...@gmail.com> >> wrote: >> >> >I suspect there's a theorem about entire complex functions f which have the property that the absolute value of f(z) tends to infinity as the absolute value of z tends to infinity. >> > What does this theorem say? I don't know of any such functions besides polynomials of degree >= 1. Is it the case that the set of functions which have this property is >> > just the set of polynomials of degree >= 1. >> >> Yes. >> >> Nonelementary proof: Look up the Piicard theorems. This is immediate >> even from the "Little" Picard theorem. >> >> Elementary proof: Let g = 1/f. Since f has only finitely many zeroes, >> g is entire except for finitely many poles. Let R be a rational >> function with the same poles as g, and with the same principal >> part at each pole. Then g  R is an entire function that tends >> to 0 at infinity, so g = R. > >Ok, I see why gR is entire but not why it tends to 0 >at infinity. What am I missing?
That follows from what I meant, not quite from what I said. Replace "Let R be a rational function with the same poles as g, and with the same principalpart at each pole" with "Let R be the rational function with the same poles as g, and with the same principal part at each pole, constructed in the obvious way."
That is, let R be a linear combination of functions of the form
1/(zp)^n,
chosen so that R has the same poles as g, with the same principal part at each pole.
> >William Hughes >

