Abstract
In this present work we derive, evaluate and produce a table of definite integrals involving logarithmic and exponential functions. Some of the closed form solutions derived are expressed in terms of elementary or transcendental functions. A substantial part of this work is new.
Highlights
Significance StatementSome of these applications are in software engineering [1], digital processing for the purposes of pattern recognition and Wiener filtering [2], distribution of sums of independent random variables [3], application to range radar profiles of naval vessels [4], gravity effect of a buried sphere and two-dimensional horizontal circular cylinder [5], and stress distribution in an infinite wedge [6]
We apply the simultaneous contour integral method [7] to an integral in Prudnikov et al and use it to derive closed forms for a Fourier cosine transform, Mellin transform, provide formal derivations for integrals in Gradshteyn and Ryzhik originally derived by Erdéyli et al and Bierens de Haan, and present a table of definite integrals of results new and known.In this paper, we derive the definite integral given byZ ∞ e ax eimx (log(c) + ix )k + e−imx (log(c) − ix )k dx (1)(e ax + 1)2 and used it to achieve several objectives
We provide formal derivations of existing formula, which is a useful exercise for determining if these formula are correct
Summary
Some of these applications are in software engineering [1], digital processing for the purposes of pattern recognition and Wiener filtering [2], distribution of sums of independent random variables [3], application to range radar profiles of naval vessels [4], gravity effect of a buried sphere and two-dimensional horizontal circular cylinder [5], and stress distribution in an infinite wedge [6] The usage of these transforms is vast and covers all aspects of everyday life.
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