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   calculating binomial sum using complex integration
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   Author  Topic: calculating binomial sum using complex integration  (Read 8356 times)
Tom27
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calculating binomial sum using complex integration  
« on: Oct 17th, 2011, 4:44am »
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hi everyone!
 
i tried to evaluate the sum
 
(n goes from 0 to infinity) C(2n,n) *x^n
 
(sorry about the mess, i could not download the math script)
 
anyway, to do that we can use the fact that  
C(2n,n) (the combinatorical number) can be written as the integral over a closed curve around the origin, of [(1+w)^(2n)]/w^(n+1) multiplied by 1/2*pi*i
 
(this is the Cauchy formula for the coefficient of z^n in the taylor series of (1+z)^2n which is also C(2n,n).
 
Now, if we take a curve on which the convergence is uniform , one can exchange summation and integration to get integral of geometrical series, that can be evaluated using residue theorem.
 
i'm kind of stuck at this stage (of finding the curve and so on..), so i would like someone to show how it's done.
 
 Grin thanks for helping!  
 
By the way, can one use LATEX here?
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