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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 =\sqrt{1-z^2})
and consider a smooth (differentiable) map ))
where 
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:
} f(z)dz=0)
