In integral calculus, Glasser's master theorem explains how a certain broad class of substitutions can simplify certain integrals over the whole interval from to It is applicable in cases where the integrals 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.[1]
A special case: the Cauchy–Schlömilch transformationedit
A special case called the Cauchy–Schlömilch substitution or Cauchy–Schlömilch transformation[2] was known to Cauchy in the early 19th century.[3] It states that if
then
where PV denotes the Cauchy principal value.
The master theoremedit
If , , and are real numbers and
then
Examplesedit
Referencesedit
^Glasser, M. L. "A Remarkable Property of Definite Integrals." Mathematics of Computation 40, 561–563, 1983.
^T. 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
^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
glasser, master, theorem, integral, calculus, explains, certain, broad, class, substitutions, simplify, certain, integrals, over, whole, interval, from, displaystyle, infty, displaystyle, infty, applicable, cases, where, integrals, must, construed, cauchy, pri. In integral calculus Glasser s master theorem explains how a certain broad class of substitutions can simplify certain integrals over the whole interval from displaystyle infty to displaystyle infty It is applicable in cases where the integrals 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 1 Contents 1 A special case the Cauchy Schlomilch transformation 2 The master theorem 3 Examples 4 References 5 External linksA special case the Cauchy Schlomilch transformation editA special case called the Cauchy Schlomilch substitution or Cauchy Schlomilch transformation 2 was known to Cauchy in the early 19th century 3 It states that if u x 1 x displaystyle u x frac 1 x nbsp then PV F u d x PV F x d x Note F u d x not F u d u displaystyle operatorname PV int infty infty F u dx operatorname PV int infty infty F x dx qquad text Note F u dx text not F u du nbsp where PV denotes the Cauchy principal value The master theorem editIf a displaystyle a nbsp a i displaystyle a i nbsp and b i displaystyle b i nbsp are real numbers and u x a n 1 N a n x b n displaystyle u x a sum n 1 N frac a n x b n nbsp then PV F u d x PV F x d x displaystyle operatorname PV int infty infty F u dx operatorname PV int infty infty F x dx nbsp Examples edit x 2 d x x 4 1 d x x 1 x 2 2 d x x 2 2 p 2 displaystyle int infty infty frac x 2 dx x 4 1 int infty infty frac dx left x frac 1 x right 2 2 int infty infty frac dx x 2 2 frac pi sqrt 2 nbsp References edit Glasser M L A Remarkable Property of Definite Integrals Mathematics of Computation 40 561 563 1983 T Amdeberhnan M L Glasser M C Jones V H Moll R Posey and D Varela The Cauchy Schlomilch transformation arxiv org pdf 1004 2445 pdf 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 1823External links editWeisstein Eric W Glasser s Master Theorem MathWorld Retrieved from https en wikipedia org w index php title Glasser 27s master theorem amp oldid 860128524, wikipedia, wiki, book, books, library,