The Cauchy-Schlomilch transformation

TL;DR

The paper discusses the Cauchy-Schlomilch transformation, an elementary integral identity, and demonstrates its utility in evaluating complex integrals and applications to probability distributions.

Contribution

It introduces and explores the Cauchy-Schlomilch transformation, showing its effectiveness in solving non-elementary integrals and applications in probability theory.

Findings

The transformation simplifies evaluation of complex integrals.

It can be used to derive new integral formulas.

Applications to probability distributions are demonstrated.

Abstract

The Cauchy-Schl\"omilch transformation states that for a function $f$ and $a, \, b > 0$, the integral of $f(x^{2})$ and $af((ax-bx^{-1})^{2}$ over the interval $[0, \infty)$ are the same. This elementary result is used to evaluate many non-elementary definite integrals, most of which cannot be obtained by symbolic packages. Applications to probability distributions is also given.