
Re: Elementary complex analysis
Posted:
Mar 7, 2013 1:23 PM


On Thu, 07 Mar 2013 17:39:14 +0100, AP <marc.pichereau@wanadoo.fr.invalid> wrote:
>On Wed, 06 Mar 2013 09:42:08 0600, David C. Ullrich ><ullrich@math.okstate.edu> wrote: > >>On Tue, 5 Mar 2013 10:29:07 0800 (PST), Paul <pepstein5@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. > in my book the little Picard theorem is : >if f is entire and if C\f(C) has , at least, two points , then f is >constant >but, I don't see how to use it >can you gives me an idea? thanks
Erm,... Ok, forget Little Picard. It's immediate from the Big Picard theorem. (Noting that if f is entire and not a polynomial then f has an essential singularity at infinity.)
>>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. >> >>Hence f = 1/R. So f is rational. Since f is also entire, f >>is a polynomial. >> >>>Thank you. >>> >>>Paul Epstein

