Logarithmic Integrals: A Review from Gradshteyn and Ryzhik to Recent Times

Abstract

The need to evaluate Logarithmic integrals is ubiquitous in essentially all quantitative areas including mathematical sciences, physical sciences. Some recent developments in Physics namely Feynman diagrams deals with the evaluation of complicated integrals involving logarithmic functions. This work deals with a systematic review of logarithmic integrals starting from $\textit{Malmsten}$ integrals to classical collection of Integrals, Series and Products by $\textit{I. S. Gradshteyn}$ and $\textit{I. M. Ryzhik}$ to recent times. The evaluation of these types of integrals involves higher transcendental functions (i.e., Hurwitz Zeta function, Polylogarithms, Lerch Transcendental, Orthogonal Polynomials, PolyGamma functions).