The Cauchy–Schlömilch transformation

Amdeberhan, T.; Glasser, M. L.; Jones, M. C.; Moll, V.; Posey, R. and Varela, D. (2019). The Cauchy–Schlömilch transformation. Integral Transforms and Special Functions, 30(12) pp. 940–961.

DOI: https://doi.org/10.1080/10652469.2019.1637432

Abstract

The Cauchy–Schlömilch transformation states that for a function f and a, b>0, the integral of ƒ(ᵡ²) and aƒ((aᵡ−bᵡ⁻¹)²) over the interval [0,∞) 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 are also given.

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