Date: Mar 8, 2013 3:55 AM
Subject: Re: Elementary complex analysis
William Hughes wrote:
>David C. Ullrich wrote:
>> >I suspect there's a theorem about entire complex function
>> >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.
>> 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?
I think the following variation of David Ullrich's argument
repairs the flaw.
Let R be the rational function consisting of the sum of all
the principal parts of g at its poles.
Then R approaches to 0 at infinity, hence since g also
approaches 0 at infinity, so does g - R.
As in David's argument, g - R is entire, hence, since g - R
appoaches 0 at infinity, it follows that g = R.
Write R = p/q as a quotient of polynomials where p,q have
no common zeros.
Then f = 1/g = 1/R = q/p.
Since f is entire, p has no zeros, hence p is a nonzero
Therefore f is a polynomial, as was to be shown.