Drexel dragonThe Math ForumDonate to the Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
William Hughes

Posts: 2,142
Registered: 12/7/10
Re: Elementary complex analysis
Posted: Mar 7, 2013 6:43 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum 1994-2015. All Rights Reserved.