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 C and π.
Highlights
This method involves using a form of Equation (1) multiplies both sides by a function, takes a definite integral of both sides
We have derived a method for expressing definite integrals in terms of special functions using contour integration
The contour we used was specific to solving integral representations in terms of the Lerch function
Summary
Table in Gradshteyn and Ryzhik: Derivation of Definite Integrals of a Hyperbolic Function. 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) multiplies both sides by a function, takes a definite integral of both sides. This yields a definite integral in terms of a contour integral.
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