AP
Posts:
137
Registered:
3/4/09


Re: Elementary complex analysis
Posted:
Mar 7, 2013 11:39 AM


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 >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

