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Topic: Summation of Series using Residue (Read 9345 times) 

knightfischer
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Summation of Series using Residue
« on: Sep 17^{th}, 2007, 4:28pm » 
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In the proof that Sum (n=inf to +inf) of {f(n)} = {sum residues pi(cot(pi(z)))f(z) at poles of f(z)}, the square Cn, with verices at (N+1/2)(1+i) is used as a path. The proof for f(z) with a finite number of poles is straightforward using the residue theorem. However, it is not clear to me how to extend this to f(z) with infinite number of poles. The book says "we can obtain the required result by an appropriate limiting procedure." With f(z) having a finite number of poles, Cn is chosen with N large enough to enclose all the poles. If f(z) has an infinite number of poles, there is no N large enough. How can the residue theorem be applied? Any help would be appreciated.


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knightfischer
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Re: Summation of Series using Residue
« Reply #1 on: Sep 17^{th}, 2007, 4:30pm » 
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The assumptions are f(z)<=M/z^k, k>1, M constants independent of N.


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Michael Dagg
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Re: Summation of Series using Residue
« Reply #2 on: Sep 29^{th}, 2007, 7:25pm » 
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Did you solve this problem?


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knightfischer
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Re: Summation of Series using Residue
« Reply #3 on: Sep 30^{th}, 2007, 7:56am » 
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No. I could not figure it out and no one seemed interested in replying. Can you help?


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Obob
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Re: Summation of Series using Residue
« Reply #4 on: Sep 30^{th}, 2007, 12:14pm » 
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I'm not really clear on what your assumption is in the second post. The assumption f(z)<=M/z^k implies that f has only a single pole.


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Sameer
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Re: Summation of Series using Residue
« Reply #5 on: Sep 30^{th}, 2007, 1:08pm » 
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I don't understand the wordings either, but for a square area and infinite poles don't you take the integrals on the sides and then let the limit of N go to infinity? That should take care of your infinite poles.


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knightfischer
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Re: Summation of Series using Residue
« Reply #6 on: Oct 1^{st}, 2007, 4:15am » 
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Sorry I was not clear. The assumptions are: along the path Cn, f(z)<=M/z^k, where k>1 and M are constants independent of N.


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knightfischer
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Re: Summation of Series using Residue
« Reply #7 on: Oct 1^{st}, 2007, 8:40am » 
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If there are an infinite number of poles, then for any N, there could still be poles outside of the square Cn. If N goes to infinity, can you still assume at some point Cn encloses all the poles, where there are an infinite number of poles, any one of which could be an infinite distance from the origin? It is not clear that the Residue Theorem can be applied unless you can prove that Cn encloses all the poles.


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