In integral calculus, Glasser's master theorem explains how a certain broad class of substitutions can simplify certain integrals over the whole real line from to The integrals in question must be construed as Cauchy principal values, and a fortiori it is applicable when the integral converges absolutely. It is named after M. L. Glasser, who introduced it in 1983.Glasser, M. L. "A Remarkable Property of Definite Integrals." Mathematics of Computation 40, 561–563, 1983.
A special case: the Cauchy–Schlömilch transformation A special case called the Cauchy–Schlömilch substitution or Cauchy–Schlömilch transformationT. Amdeberhnan, M. L. Glasser, M. C. Jones, V. H. Moll, R. Posey, and D. Varela, "The Cauchy–Schlömilch transformation", arxiv.org/pdf/1004.2445.pdf was known to Cauchy in the early 19th century.A. L. Cauchy, "Sur une formule generale relative a la transformation des integrales simples prises entre les limites 0 et ∞ de la variable." Oeuvres completes, serie 2, Journal de l’ecole Polytechnique, XIX cahier, tome XIII, 516–519, 1:275–357, 1823 It states that for ,
where PV denotes the Cauchy principal value and is a function which is integrable on the real line at least in the sense of the Cauchy principal value.
The master theorem
If , , and are real numbers and
then
Examples
where the first equality comes from cancelling , the second from Cauchy–Schlömilch, and the last one from a substitution and the integral of the arctangent function.
References
External links