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Topic: the Extended Hurwitz's Theorem (Read 11324 times) |
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immanuel78
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the Extended Hurwitz's Theorem
« on: Oct 26th, 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 |z-a|=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 |z-a|=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?
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| « Last Edit: Oct 26th, 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 3rd, 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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Regards,
Michael Dagg
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immanuel78
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Re: the Extended Hurwitz's Theorem
« Reply #2 on: Nov 29th, 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.
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| « Last Edit: Nov 29th, 2008, 5:31am by immanuel78 » |
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