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Table in Gradshteyn and Ryzhik: Derivation of definite integrals of a Hyperbolic Function
Version 1
: Received: 8 May 2021 / Approved: 10 May 2021 / Online: 10 May 2021 (13:51:04 CEST)
A peer-reviewed article of this Preprint also exists.
Reynolds, R.; Stauffer, A. Table in Gradshteyn and Ryzhik: Derivation of Definite Integrals of a Hyperbolic Function. Sci 2021, 3, 37. Reynolds, R.; Stauffer, A. Table in Gradshteyn and Ryzhik: Derivation of Definite Integrals of a Hyperbolic Function. Sci 2021, 3, 37.
Abstract
We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage of using special functions is their analytic continuation which widens the range of the parameters of the definite integral over which the formula is valid. We give as examples definite integrals of logarithmic functions times a trigonometric function. In various cases these generalizations evaluate to known mathematical constants such as Catalan’s constant and π
Keywords
entries in Gradshteyn and Rhyzik, Lerch function, Logarithm function, Contour Integral, Cauchy, Infinite Integral
Subject
Computer Science and Mathematics, Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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