AP
Posts:
134
Registered:
3/4/09
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Re: Elementary complex analysis
Posted:
Mar 7, 2013 11:39 AM
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On Wed, 06 Mar 2013 09:42:08 -0600, David C. Ullrich
<ullrich@math.okstate.edu> 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.
>
>Non-elementary proof: Look up the Piicard theorems. This is immediate
>even from the "Little" Picard theorem.
in my book the little Picard theorem is :
if f is entire and if C\f(C) has , at least, two points , then f is
constant
but, I don't see how to use it
can you gives me an idea? thanks
>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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