## Factorial series

Evaluate the series

Solution

Let denote Kronecker’s delta and . We have successively,

It follows that

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.

## Square logarithmic integral

Prove that

Solution

Lemma 1: Let . It holds that

Lemma 2: Let . It holds that

We begin by squaring the identity of lemma 2. Hence,

Integrating the last equation we get,

Expanding the LHS we get that

Finally,

## Otto Dunkel Memomorial

Prove that

Solution

We have successively:

since for it holds that

## A squared trigamma series

Prove that

Solution

Since we have successively: