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February 14th, 2012, 04:38 PM   #1
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Gamma Function

Show is holomorphic in the right half plane by the following three steps:

1) Let and show the function is holomorphic in each strip.

2) Let and show is holomorphic in

3) Show that as we have uniformly on the compact subsets of the strip by obtaining an estimate
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February 15th, 2012, 04:13 PM   #2
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Re: Gamma Function

[color=#000000]Here is a simpler proof that might help you with your problem.

Since the given integral is improper, we split the integral in the following way .

The integral converges absolutely for , for every bounded and closed region in , if for we have



and since converges, by Weierstra?' criterion follows the uniform convergence of the integral on . So we deduce that the integral converges uniformly and absolutely on every bounded region . Now if is in the disc , for ,



using again the Weierstra? criterion we deduce (since converges), the integral converges uniformly and absolutely on every bounded region . So ? is regular in .[/color]
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