Evaluating residues and integrals through Negative Dimensional Integration Method (NDIM)

TL;DR

This paper introduces the Negative Dimensional Integration Method (NDIM) as an alternative to the residue theorem for evaluating residues and real integrals, simplifying calculations by avoiding pole determination.

Contribution

The paper presents NDIM as a novel approach for integral evaluation, eliminating the need for pole analysis and enabling simultaneous results for various pole orders.

Findings

NDIM simplifies residue and integral calculations.

The method requires only Gaussian integration and linear algebra.

It provides multiple results for different pole orders simultaneously.

Abstract

The standard way of evaluating residues and some real integrals through the residue theorem (Cauchy's theorem) is well-known and widely applied in many branches of Physics. Herein we present an alternative technique based on the negative dimensional integration method (NDIM) originally developed to handle Feynman integrals. The advantage of this new technique is that we need only to apply Gaussian integration and solve systems of linear algebraic equations, with no need to determine the poles themselves or their residues, as well as obtaining a whole class of results for differing orders of poles simultaneously.