DEFINITE INTEGRAL OF LOGARITHMIC FUNCTIONS AND POWERS IN TERMS OF THE LERCH FUNCTION

A family of generalized definite logarithmic integrals given by $$\int_{0}^{1}\frac{\left(x^{ i m} (\log (a)+i \log (x))^k+x^{-i (a)-i (x))^k\right)}{(x+1)^2}dx$$built over the Lerch function has its analytic properties and special values listed in explicit detail. We use general method as [6] to derive this integral. then give a number examples that can be derived from integral terms well known functions.