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Math Help - Residue of Gamma

  1. #1
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    Question Residue of Gamma

    Can anyone offer advice on how one might use the residue of gamma to show the integral gamma (z) dz=-4piei/3 where C=(z:z[Z+2-i]=2).......or am I on the right track at all?
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  2. #2
    MHF Contributor chisigma's Avatar
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    Two questions...

    a) by 'gamma function' do you mean...

    \Gamma (z) = \int_{0}^{\infty} t^{z-1} e^{-t} dt

    ... or some else?...

    c) which is the path in the complex plane along which you intend to integrate the 'gamma function'?...

    Kind regards

    \chi \sigma
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  3. #3
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    Quote Originally Posted by pennylane View Post
    Can anyone offer advice on how one might use the residue of gamma to show the integral gamma (z) dz=-4piei/3 where C=(z:z[Z+2-i]=2).......or am I on the right track at all?
    It says here that the residue of the gamma function at the simple pole z=–n is (-1)^n/n!\,. If C is meant to be the circle of radius 2 centred at –2+i then the poles inside the circle are at –1, –2 and –3, and 2\pi i times the sum of the residues given by that formula is indeed -4\pi i/3.
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  4. #4
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    Ah yes it is indeed the Res(gamma,-k)=(-1)^k/k! as per the link........excellent.....thank you.
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