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

Robert Reynolds     (Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Canada)
Allan Stauffer     (Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Canada)

Abstract


A family of generalized definite logarithmic integrals given by $$
\int_{0}^{1}\frac{\left(x^{ i m} (\log (a)+i \log (x))^k+x^{-i m} (\log (a)-i \log (x))^k\right)}{(x+1)^2}dx$$
built over the Lerch function has its analytic properties and special values listed in explicit detail. We use 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.


Keywords


Entries of Gradshteyn and Ryzhik, Lerch function, Knuth's Series

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References


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DOI: http://dx.doi.org/10.15826/umj.2021.1.008

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