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

TL;DR

This paper introduces a contour integration method to derive definite integrals involving hyperbolic functions, expanding their applicability via special functions and illustrating with examples that connect to known constants.

Contribution

It presents a novel contour integration approach for deriving definite integrals and their sums, emphasizing the use of special functions for broader parameter ranges.

Findings

Derived new integral formulas involving hyperbolic functions.

Connected integrals to mathematical constants like Catalan's constant and pi.

Demonstrated the method with examples involving logarithmic and trigonometric functions.

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 $\pi$.