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. > > Non-elementary 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 g-R is entire but not why it tends to 0 at infinity. What am I missing?