Abstract
This manuscript concerns two definite integrals that could be connected to the Bose-Einstein and the Fermi-Dirac functions in the integrands, separately, with numerators slightly modified with a difference in two expressions that contain the Fourier kernel multiplied by a polynomial and its complex conjugate. In this work, we use our contour integral method to derive these definite integrals, which are given by ∫0∞ie−imx(log(a)−ix)k−eimx(log(a)+ix)k2eαx−1dx and ∫0∞ie−imx(log(a)−ix)k−eimx(log(a)+ix)k2eαx+1dx in terms of the Lerch function. We use these two definite integrals to derive formulae by Erdéyli and Watson. We derive special cases of these integrals in terms of special functions not found in current literature. Special functions have the property of analytic continuation, which widens the range of computation of the variables involved.
Highlights
In 1953, Erdélyi et al [1] and Watson et al [2] published famous books containing a significant number of definite integrals not containing formal derivations
We focus on deriving a few of these integrals, which have interesting closed forms, including deriving new Mellin transforms in terms of special functions
Mellin transforms [3] are used in solving the potential problem in a wedge-shaped region and solving linear differential equations in electrical engineering
Summary
In 1953, Erdélyi et al [1] and Watson et al [2] published famous books containing a significant number of definite integrals not containing formal derivations. In this manuscript, we focus on deriving a few of these integrals, which have interesting closed forms, including deriving new Mellin transforms in terms of special functions. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations This method involves using a form of Equation (1), multipliying both sides by a function, taking a definite integral of both sides. We multiply both sides of Equation (1) by another function and take the infinite sum of both sides such that the contour integral of both equations are the same
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