Definite Integral of Logarithmic Functions and Powers in Terms of the Lerch Function

Abstract

A family of generalized definite logarithmic integrals given by ∫10(xim(log(a)+ilog(x))k+x−im(log(a)−ilog(x))k)(x+1)2dx built over the Lerch function has its analytic properties and special values listed in explicit detail. We use the general method as given in [5] to derive this integral. We then give a number of examples that can be derived from the general integral in terms of well known functions.