Meromorphic Reduction in Integration

TL;DR

This paper discusses a method for transforming integrals involving meromorphic functions into equivalent integrals with simpler kernels, using properties of analytic functions and principal value convergence.

Contribution

It introduces a new approach for reducing integrals with meromorphic functions to simpler forms through analytic and principal value techniques.

Findings

Established conditions for integral equivalence involving meromorphic functions.

Derived explicit formulas for transformed integral kernels.

Demonstrated the method's applicability to specific classes of functions.

Abstract

It is argued that for certain meromorphic functions $u:\cal{R}\rightarrow\cal{R}$ and analytic function $ A_1$ and for any integrable function $F$, as long as it converges as a Cauchy Principal Value,, $$\int_{-\infty}^{\infty}A_1(x)F[u(x)] dx=\int_{-\infty}^{\infty} A_2(x)F(x) dx,$$ where $A_2$ is also analytic.