Paul
Posts:
572
Registered:
7/12/10


Re: Elementary complex analysis
Posted:
Mar 6, 2013 11:43 AM


On Wednesday, March 6, 2013 3:42:08 PM UTC, David C. Ullrich 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. > > > > 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
Great reply. Thanks!
Paul

