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Thread: contour integral around $\infty$

  1. #1
    Nov 2012

    contour integral around $\infty$

    Consider the function


    where a,b\in D(0,r) , the disc of radius r centered at the origin, open, and r>0. Show that f is holomophic in an open subset of complex plane, containing the complement of the disc: \mathbb C-D(0,r) and compute the integral


    where n\geq 0 is an integer.

    I'm quite confused about this problem. Because, the $Log$ in the RHS should be the principal branch, i.e. Log(z)=log|z|+iArg(z), with Arg(z)\in (-\pi,\pi) and Arg(0)=0. Then i should prove that \frac{z-a}{z-b} does not intersect the real negative axis, but how to do this? However, suppose i could prove the holomorphicity, then i'm left to compute the integral, for which i tought to use the residue formula applied to \infty point, so calling I the integral,

    I=2\pi i Ind(\gamma,\infty)Res(f,\infty)

    where \gamma is the contour |z|=1, so that Ind(\gamma,\infty)=-1. But then the computation of residue is very hard for me. Some help?
    Last edited by tenderline; Nov 29th 2012 at 07:46 AM.
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