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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:
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.
Let be the roots of the polynomial
Suppose the roots of polynomial are where
Let . Then, the are the zeros of in the disk where is chosen such that for .
Jensen’s inequality implies that
Applying Cauchy – Schwartz yields,
Letting and we get the result.
Let be analytic in the disk . Prove that:
It follows from Taylor that and the convergence is uniform. Hence,
We have that and for we also have that due to
So if then lies within the disk ; hence the integral equals whereas if then lies outside the disk ; hence
Let denote one of the Jacobi Theta functions. Prove that
We have successively,
The sum is evaluated as follows. Consider the function
and integrate it around a square with vertices . The function has poles at every integer with residue as well as at with residues . We also note that as the contour integral of tends to . Thus,
and the exercise is complete.
Consider the points and . Let be a point of the plane such that . Set to be the angle that is defined by and . ( the one that is less than .) Prove that the function is harmonic.
Solution [by Demetres Skouteris]