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# Tag Archives: Complex Analysis

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.

## Upper bound of max product

Let be the roots of the polynomial

Prove that:

Solution

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,

Therefore,

Letting and we get the result.

## Contour integral

Let be analytic in the disk . Prove that:

Solution

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

## Integral of Jacobi Theta function

Let denote one of the Jacobi Theta functions. Prove that

Solution

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,

Hence,

and the exercise is complete.

## Harmonic function

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]

The complex function has a holomorphic branch in the half plane and its imaginary part is the desired angle. Hence, the function of the angle is harmonic.