An Extension of the Method of Brackets. Part 1

TL;DR

This paper extends the method of brackets for evaluating definite integrals by incorporating concepts of null and divergent series, enabling the evaluation of more complex integrals using formal series representations.

Contribution

It introduces a novel extension of the method of brackets utilizing null and divergent series concepts for broader integral evaluation capabilities.

Findings

Successfully evaluated classical integrals from Gradshteyn and Ryzhik

Extended the method of brackets to handle divergent series

Demonstrated the effectiveness of the extension with various examples

Abstract

The method of brackets is an efficient method for the evaluation of a large class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the concepts of null and divergent series. These are formal representations of functions, whose coefficients $a_{n}$ have meromorphic representations for $n \in \mathbb{C}$, but might vanish or blow up when $n \in \mathbb{N}$. These ideas are illustrated with the evaluation of a variety of entries from the classical table of integrals by Gradshteyn and Ryzhik.