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

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Paul

Posts: 411
Registered: 7/12/10
Re: Elementary complex analysis
Posted: Mar 6, 2013 11:43 AM
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On Wednesday, March 6, 2013 3:42:08 PM UTC, David C. Ullrich wrote:
> On Tue, 5 Mar 2013 10:29:07 -0800 (PST), Paul <pepstein5@gmail.com>
>
> wrote:
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>
>

> >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.
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> > 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
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> > just the set of polynomials of degree >= 1.
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>
>
>
>
> Yes.
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>
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> Non-elementary proof: Look up the Piicard theorems. This is immediate
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> even from the "Little" Picard theorem.
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>
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> Elementary proof: Let g = 1/f. Since f has only finitely many zeroes,
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> g is entire except for finitely many poles. Let R be a rational
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> function with the same poles as g, and with the same principal
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> part at each pole. Then g - R is an entire function that tends
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> to 0 at infinity, so g = R.
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>
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> Hence f = 1/R. So f is rational. Since f is also entire, f
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> is a polynomial.
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>
>

> >Thank you.
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> >
>
> >Paul Epstein

Great reply. Thanks!

Paul



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