The Cauchy–Schlömilch transformation. Issue 12 (2nd December 2019)

• Record Type: Journal Article
• Authors: Amdeberhan, T.; Glasser, M. L.; Jones, M. C.; Moll, V.; Posey, R.; Varela, D.
• Is Part Of: Integral transforms and special functions. Volume 30:Issue 12(2019)
• Page Start: 940
• Page End: 961
• Publication Date: 2019-12-02
• Subjects: Definite integrals -- transformations -- Laplace integral -- error function -- elliptic integrals -- Bessel function -- Riemann zeta function; Primary: 33B20 -- Secondary: 33E05 -- 60E05; Integral transforms -- Periodicals; Transcendental functions -- Periodicals; Transformations (Mathematics) -- Periodicals; Calculus, Integral -- Periodicals; 515
• DOI: 10.1080/10652469.2019.1637432
• Languages: English
• ISSNs: 1065-2469

Abstract

ABSTRACT: The Cauchy–Schlömilch transformation states that for a function f and a, b > 0, the integral of f ( x 2 ) and a f ( ( a x − b x − 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.