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Topic: the Extended Hurwitz's Theorem (Read 10816 times) 

immanuel78
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the Extended Hurwitz's Theorem
« on: Oct 26^{th}, 2008, 7:49am » 
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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?

« Last Edit: Oct 26^{th}, 2008, 10:10pm by immanuel78 » 
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Michael Dagg
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Re: the Extended Hurwitz's Theorem
« Reply #1 on: Nov 3^{rd}, 2008, 8:09pm » 
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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?).


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immanuel78
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Re: the Extended Hurwitz's Theorem
« Reply #2 on: Nov 29^{th}, 2008, 5:26am » 
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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.

« Last Edit: Nov 29^{th}, 2008, 5:31am by immanuel78 » 
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