Integral representations with logarithmic and certain special functions

In this work we used contour integration to derive a Mellin transform involving the product of the logarithmic function raised to complex number power and rational functions. This transform is expressed in terms of a series involving the Hurwitz-Lerch zeta function \(\Phi(z,s,a)\). Special cases of this transform are in terms of definite integrals involving Malmsten logarithm forms, reciprocal logarithm forms and Kölbig type integrals in terms of special functions and fundamental constants. We also present errata for some integrals in the books [I. S. Gradshteyn, I. M. Ryzhik, Elsevier/Academic Press, Amsterdam, (2015)] and [D. Bierens De Haan, C. G. van der Post: Amsterdam, The Netherlands, (1867)]. In this paper we first discuss the applications of a particular definite integral, then we produce the correct version of the definite integral in [I. S. Gradshteyn, I. M. Ryzhik, Elsevier/Academic Press, Amsterdam, (2015)]. Next we write down the contour integral representation for this definite integral, followed by using a contour integral method to derive a new definite integral involving the logarithmic function expressed in terms infinite series of a special function. We then evaluate this definite integral in terms of known functions and constants and end with a conclusion about the paper with future work.