

Title: the Extended Hurwitz's Theorem Post by immanuel78 on Oct 26^{th}, 2008, 7:49am The Hurwitz's theorem that I have seen in the textbook until now is as follows : H(G) = the set of analytic functions in G M(G) = the set of meromorphic functions in G C = the set of complex numbers C infinite = C union {infinite} Let {fn} be a sequence in H(G) and fn > f , where f:G>C is continuous. If f is identically not zero, closed disk B(a;R) in G and f(z) not zero in za=R , then there in an integer N such that for n>= N, f and fn have the same number of zeros in open disk B(a;R). Now I think that {fn} in H(G) can be extended to {fn} in M(G). That is, Let {fn} in M(G) and fn > f , where f : G > C infinite is continuous If f is identically not zero or infinite, closed disk B(a;R) in G and f(z) not zero or infinite in za=R , then there in an integer N such that for n>= N, [the number of zeros of f in B(a;R)]  [the number of poles of f in B(a;R)] = [the number of zeros of fn in B(a;R)]  [the number of poles of fn in B(a;R)]. Have you seen this extened theorem in your textbook or exercises? 

Title: Re: the Extended Hurwitz's Theorem Post by Michael Dagg on Nov 3^{rd}, 2008, 8:09pm Start with p. 153 in Conway's book. I believe the problem you mention before regarding Hardy's Theorem came form Conway (actually its problem from the book?). 

Title: Re: the Extended Hurwitz's Theorem Post by immanuel78 on Nov 29^{th}, 2008, 5:26am Theorem I mentioned above is correct, but I came to know actually a stronger result is true. There is an integer N such that for n>=N, the number of zeroes of fn in B(a;R) equals the number of zeroes of f in B(a;R), and also the number of poles of fn in B(a;R) equals the number of poles of f in B(a;R). A statement and a proof of Hurwitz's theorem for meromorphic functions can be found in the book Complex Function Theory by Maurice Heins (Academic Press, 1968 ), Theorem 4.4 on page 180. 

Powered by YaBB 1 Gold  SP 1.4! Forum software copyright © 20002004 Yet another Bulletin Board 