# A Method for Evaluating Definite Integrals in terms of Special Functions with Examples

## TL;DR

This paper introduces a contour integration method to evaluate definite integrals and infinite sums expressed as special functions, enhancing the range of valid parameters through analytic continuation and illustrating with examples involving logarithmic and trigonometric functions.

## Contribution

It provides a novel contour integration approach for deriving definite integrals in terms of special functions, with proofs and practical examples.

## Key findings

- Evaluates integrals involving logarithmic and trigonometric functions.
- Expresses integrals as special functions with extended parameter ranges.
- Connects results to known constants like Catalan's constant and pi.

## 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 $\pi$.