William Hughes wrote: >David C. Ullrich wrote: >>Paul 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. >> >> 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?
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 constant.