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# Tag Archives: Complex Analysis

## Contour radical integral

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]