Negative dimensional approach to evaluating real integrals

TL;DR

This paper introduces a novel method using negative dimensional integration to evaluate real integrals arising in differential equations, providing an alternative to traditional residue calculus in complex analysis.

Contribution

It presents the first application of negative dimensional integration to evaluate real integrals in the context of differential equations, offering a new analytical tool.

Findings

Successfully applied negative dimensional approach to a specific integral example.

Provided an alternative to residue theorem for evaluating integrals in physics problems.

Demonstrated the method's potential for broader applications in mathematical physics.

Abstract

In solving the differential equation for a non damped harmonic oscillator one meets, after subjecting the equation to a Fourier transformation, an integration in the complex $\omega$ plane. In most cases such an integral is evaluated by calculating residues together with some physical input such as the principle of causality to define which pole residues are relevant to the physical problem. For this kind of application, Cauchy's theorem or residue theorem can be applied to evaluate certain real integrals. Here we present an alternative approach based on the concept of negative dimensional integration to treat such integrals and give an specific example on how this is accomplished.