On Tue, 5 Mar 2013 10:29:07 -0800 (PST), Paul <email@example.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.
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.
Hence f = 1/R. So f is rational. Since f is also entire, f is a polynomial.