Abstract
It is always useful to improve the catalogue of definite integrals available in tables. In this paper we use our previous work on Lobachevsky integrals to derive entries in the tables by Bierens De Haan and Anatolli Prudnikov featuring errata results and new integral formula for interested readers. In this work we derive a definite integral given by (1) in terms of the Lerch function. The importance of this work lies in the derivation of known and new results not presently found in current literature. We used our contour integral method and applied it to an integral in Prudnikov and used it to derive a closed form solution in terms of a special function. The advantage of using a special function is the added benefit of analytic continuation which widens the range of computation of the parameters. Special functions have significance in mathematical analysis, functional analysis, geometry, physics, and other applications. Special functions are used in the solutions of differential equations or integrals of elementary functions. Special functions are linked to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics.
Highlights
In 1867 David Bierens De Haan [3] and 1990 Anatolii Prudnikov [5] both produced famous books on definite integrals
The authors expand on their work in [7] and their contour integral method and applied it to an interesting integral in the book of Prudnikov et al [5] and expressed its closed form in terms of the Lerch function
The Lerch function being a special function has the fundamental property of analytic continuation, which enables us to widen the range of evaluation for the parameters involved in our definite integral
Summary
In 1867 David Bierens De Haan [3] and 1990 Anatolii Prudnikov [5] both produced famous books on definite integrals. The authors expand on their work in [7] and their contour integral method and applied it to an interesting integral in the book of Prudnikov et al [5] and expressed its closed form in terms of the Lerch function. Where C is in general an open contour in the complex plane where the bilinear concomitant has the same value at the end points of the contour This method involves using a form of equation (3) multiply both sides by a function, take a definite integral of both sides. A second contour integral is derived by multiplying equation (3) by a function and performing some substitutions so that the contour integrals are the same
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