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Title: Re: Maximum Modulus Theorem Post by Eigenray on Apr 8th, 2009, 5:18pm Edit: original question (paraphrase) : Suppose fn(z) is a sequence of analytic functions on the unit disk which converge uniformly on compact subsets to a non-zero function f. If each fn has at most m zeroes (counting multiplicity), show that f has at most m zeroes. Do you know [link=http://en.wikipedia.org/wiki/Hurwitz%27s_theorem#Hurwitz.27s_theorem_in_complex_analysis]Hurwitz's theorem[/link]? It basically comes from the argument principle. |
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Title: Re: Maximum Modulus Theorem Post by Eigenray on Apr 8th, 2009, 9:09pm on 04/08/09 at 20:00:26, trusure wrote:
Yes that's right; it's possible for some of the zeroes to wander off, as the example fk(z) = z-1+1/k shows. But they can't suddenly appear in the limit. If f had more than n zeros, we could fix a radius r < 1 within which f had more than n zeros, and eventually the fk would have the same number of zeros in that disk, a contradiction. |
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Title: Re: Maximum Modulus Theorem Post by Eigenray on Apr 11th, 2009, 12:31am on 04/10/09 at 21:29:26, trusure wrote:
Sure, just take fn(z) = zk(z - (1-1/n))m-k. Quote:
The zero function does not have zero roots! |
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Title: Re: Maximum Modulus Theorem Post by Grimbal on Apr 12th, 2009, 7:53am Hello Eigenray, Was there somebody else here, or are you having a talk with your imaginary friend? :P |
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Title: Re: Maximum Modulus Theorem Post by trusure on Apr 12th, 2009, 8:09am ohh Sorry, I don't know how the questions deleted !? maybe by a mistake I will post the problem again sorry again |
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Title: Re: Maximum Modulus Theorem Post by trusure on Apr 12th, 2009, 8:21am the problem was to find a sequnce of analytic functions as indicated above, and I was wondering that the sequence given by Mr. Eigenray has exactly m zeros inside B(0,1) not at least m zeros. |
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Title: Re: Maximum Modulus Theorem Post by Eigenray on Apr 12th, 2009, 2:14pm If it has exactly m zeros then it also has at least m zeros. I suppose you could give it m+1 zeros if you really wanted to. |
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