## 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.

## Logarithmic Gaussian integral

Let denote the Euler’s constant. Prove that:

**Solution**

First of all by making the substitution we get that

For let us consider the function

where is the Euler’s Gamma function. Differentiating once we get:

Thus, the desired integral is obtained by setting . Hence,

## Series of zeta sum

Let denote the Riemann zeta function. Evaluate the series:

**Solution**

Let . We are proving the more general result.

where denotes the digamma function.

First of all, we note that:

Thus,

due to the reflection formula .

**Side note: **If then the sum equals .

## On a log Gamma integral using Riemann sums

Evaluate the integral

using Riemann sums.

**Solution**

Partition the interval into subintervals of length . This produces,

(1)

On the other hand, assuming is even:

since it holds that

Euler’s Gamma reflection formula was used at line . Letting we get that

If is odd we work similarly.

## A beautiful limit with harmonic number

Let denote the – th harmonic number. Prove that

**Solution**

We note that

At the last integral we apply the substitution . Thus,

After all these we have that

proving the result.