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Re: Elementary complex analysis
Posted:
Mar 7, 2013 1:23 PM
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On Thu, 07 Mar 2013 17:39:14 +0100, AP
<marc.pichereau@wanadoo.fr.invalid> wrote:
>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
Erm,... Ok, forget Little Picard. It's immediate from the Big
Picard theorem. (Noting that if f is entire and not a polynomial
then f has an essential singularity at infinity.)
>>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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