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## 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.