Integrals Associated with the Digamma Integral Representation

TL;DR

This paper explores specific integrals related to the Digamma function, providing relations to express them in terms of Polygamma functions, thereby advancing integral representations of special functions.

Contribution

It introduces new relations among integrals involving x^n/(x^2+b^2)^j and exponential kernels, aiming to express them through Polygamma functions.

Findings

Derived relations reduce complex integrals to Polygamma functions.

Connected integral representations to the Digamma function.

Enhanced understanding of integral forms related to special functions.

Abstract

The definite integral with the kernel x/(x^2+b^2)/[\exp(2\pi x)-1] integrated from x=0 to infinity is the main term of a representation of the Digamma-Function psi(b), the derivative of the logarithm of the Gamma-Function. We present relations within the set of integrals over x^n/(x^2+b^2)^j/[\exp(\mu x)-1]^s for small integer exponents n, j and s with the aim to reduce them all to Polygamma-Functions.