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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

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.

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

Date Subject Author
3/5/13 Paul
3/5/13 bacle
3/5/13 W^3
3/5/13 Scott Berg
3/5/13 J. Antonio Perez M.
3/6/13 Scott Berg
3/6/13 Frederick Williams
3/6/13 Robin Chapman
3/6/13 David C. Ullrich
3/6/13 Paul
3/7/13 AP
3/7/13 David C. Ullrich
3/7/13 William Hughes
3/8/13 quasi
3/8/13 William Hughes
3/8/13 quasi
3/8/13 AP
3/8/13 David C. Ullrich
3/8/13 W^3
3/9/13 David C. Ullrich