wu :: forums (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
general >> complex analysis >> holomorphic polynomials
(Message started by: Mary I on Dec 2nd, 2005, 5:04am)

Title: holomorphic polynomials
Post by Mary I on Dec 2nd, 2005, 5:04am
Hello!

If we know that f is holomorphic on D(0,1) and assume that f^2 is a holomorphic polynomial on D(0,1), does it follow that f is also a polynomial on D(0,1)?

I am almost sure that f is a polynomial but how can I show it?
I have tried to use the Cauchy product but it didn't work out.  

:-[

Title: Re: holomorphic polynomials
Post by Eigenray on Dec 2nd, 2005, 5:23am
No.  Note that any holomorphic function which has no zeroes on a simply-connected domain like D(0,1) has a holomorphic square root there.

Title: Re: holomorphic polynomials
Post by Icarus on Dec 2nd, 2005, 3:48pm
For example, f(x) = sqrt(x2 + 4) satisfies all your conditions, but is not a polynomial.

(By the way, "holomorphic polynomial" is redundant. All polynomials are holomorphic.)



Powered by YaBB 1 Gold - SP 1.4!
Forum software copyright © 2000-2004 Yet another Bulletin Board