Meromorphic Reduction in Integration

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.