Integrals Associated with the Digamma Integral Representation

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.