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Topic: Elementary complex analysis
Replies: 19   Last Post: Mar 9, 2013 11:35 AM

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David C. Ullrich

Posts: 21,553
Registered: 12/6/04
Re: Elementary complex analysis
Posted: Mar 6, 2013 10:42 AM
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On Tue, 5 Mar 2013 10:29:07 -0800 (PST), Paul <>

>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.

>Thank you.
>Paul Epstein

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