Table in Gradshteyn and Ryzhik: Derivation of definite integrals of a Hyperbolic Function

TL;DR

The paper develops a contour-integration framework to derive definite integrals of hyperbolic functions and related infinite sums, expressing results in terms of special functions such as the Lerch transcendent, Hurwitz zeta, and log-gamma with harmonic-number corrections. It employs a generalized Cauchy integral formula and parameterized contour manipulations to obtain representations valid for broad parameter regimes via analytic continuation, linking the results to classical table entries and new results alike. The main contributions include explicit Lerch-function closed forms for a wide class of integrals, derivations of several Gradshteyn–Ryzhik and Brychkov entries, and multiple examples that reduce to well-known constants such as Catalan’s constant, π, and gamma-function expressions; numerical verification is provided. The method offers a powerful, extensible approach for obtaining closed forms of hyperbolic integrals and their associated sums, with potential to yield further entries through alternate contours and parameter choices, thereby enriching analytic tools for definite integrals in mathematical physics and analysis.

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 $π$.