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Math Help - Integral over head-light contour via Residue Theorem

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    Integral over head-light contour via Residue Theorem

    This is a challenge question that I already know how to solve: Let f(z)=\sqrt{1-z^2} and consider a smooth (differentiable) map f(H(t)) where H is the general shape shown in the first plot below (loops around the branch-points,then looping around to close the contour). That is, the contour traverses an analytic path over multiple branches of the integrand. Show via the Residue Theorem:

    \mathop\oint\limits_{H(t)} f(z)dz=0
    Attached Thumbnails Attached Thumbnails Integral over head-light contour via Residue Theorem-multiloop-1.jpg  
    Last edited by shawsend; February 6th 2010 at 01:19 PM. Reason: labeled plot
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