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Old Oct 1st 2009, 11:24 PM   #1
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theorem of residues

the theorem about sum of residues tells us that the integral f(z)dz around a closed curve in complex plane containing many singular points is equal to 2 pi i times the sum of the residues calculated at each singular point.in the proof we make cuts and apply cauchy integral theorem in each and then add.then why is it not (n)times the integral f(z)dz=2pi i times sum of resiodues.
where 'n' is the no of singular points.
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Old Oct 3rd 2009, 10:38 AM   #2
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theorem of residues

For one thing, the integrals round the non analytic points can be different to 2*pi*i for example it could be 3*pi*i for one and 0.5*pi*i for another. So although the idea is the same, you need to take the behaviour of each individual point into account using it's Laurent series I think.
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Old Oct 4th 2009, 12:13 AM   #3
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a possible solution

i think i was wrong. i thought the cuts were made one at a time with individual singular points. but the story is that all cuts are made the same time, so the the integration is a tangled mess of paths but on the brighter side, the main integration of f(z)dz comes into account only once throught the outermost contour. have i got it right?
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Old Oct 4th 2009, 01:21 AM   #4
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Why dont you post this in the math help forum also? You are bound to get a faster response there. ( im really rusty with this stuff.)
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