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Consider the branch of which is defined outside the segment and which coincides with the positive square root for . Let then evaluate the contour integral:

Solution

It is a classic case of residue at infinity. Subbing the counterclockwise contour integral rotates the northern pole of the Riemannian sphere to the southern one and the contour integral is transformed to a clockwise one. Hence:

The equality does hold for all if we take the standard branch , otherwise it is not that obvious why this holds, since we are dealing with a multi-valued function.