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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 8, 2013 10:48 AM
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On Thu, 7 Mar 2013 15:43:09 -0800 (PST), William Hughes
<wpihughes@gmail.com> wrote:

>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?


That follows from what I meant, not quite from what I said.
Replace "Let R be a rational
function with the same poles as g, and with the same principalpart at
each pole" with "Let R be the rational
function with the same poles as g, and with the same principal
part at each pole, constructed in the obvious way."

That is, let R be a linear combination of functions of the form

1/(z-p)^n,

chosen so that R has the same poles as g, with the same
principal part at each pole.

>
>William Hughes
>





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