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

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William Hughes

Posts: 1,799
Registered: 12/7/10
Re: Elementary complex analysis
Posted: Mar 7, 2013 6:43 PM
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On Mar 6, 4:42 pm, David C. Ullrich <ullr...@math.okstate.edu> wrote:
> On Tue, 5 Mar 2013 10:29:07 -0800 (PST), Paul <pepste...@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.


Ok, I see why g-R is entire but not why it tends to 0
at infinity. What am I missing?

William Hughes





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