Note on a Stieltjes Transform in terms of the Lerch Function

Abstract

In this work the authors derive the Stieltjes transform of the logarithmic function in terms of the Lerch function. This transform is used to derive closed form solutions involving fundamental constants and special functions. Specifically we derive the definite integral given by

\[
\int_{0}^{\infty} \frac{(1-b x)^m \log ^k(c (1-b x))+(b x+1)^m \log ^k(c (b x+1))}{a+x^2}dx
\]

where $a,b,c,m$ and $k$ are general complex numbers subject to the restrictions given in connection with the formulas.