Glasser's master theorem

Last updated April 04, 2019

In integral calculus, Glasser's master theorem explains how a certain broad class of substitutions can simplify certain integrals over the whole interval from $-\infty$ to $+\infty .$ 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]

In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.

In mathematics, an infinite series of numbers is said to converge absolutely if the sum of the absolute values of the summands is finite. More precisely, a real or complex series $is said to converge absolutely if for some real number . Similarly, an improper integral of a function,, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if$

A special case: the Cauchy–Schlömilch transformation

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

Baron Augustin-Louis Cauchy was a French mathematician, engineer and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors. He almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra.

u=x-{\frac {1}{x}}\,

then

\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)

where PV denotes the Cauchy principal value.

The master theorem

If $a$ , $a_{i}$ , and $b_{i}$ are real numbers and

u=x-a-\sum _{n=1}^{N}{\frac {|a_{n}|}{x-b_{n}}}

then

\operatorname {PV} \int _{-\infty }^{\infty }F(u)\,dx=\operatorname {PV} \int _{-\infty }^{\infty }F(x)\,dx.

Examples

$\int _{-\infty }^{\infty }{\frac {x^{2}\,dx}{x^{4}+1}}=\int _{-\infty }^{\infty }{\frac {dx}{\left(x-{\frac {1}{x}}\right)^{2}+2}}=\int _{-\infty }^{\infty }{\frac {dx}{x^{2}+2}}={\frac {\pi }{\sqrt {2}}}.$

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References

↑ 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

External links

Weisstein, Eric W. "Glasser's Master Theorem". MathWorld .

Eric Wolfgang Weisstein is an encyclopedist who created and maintains MathWorld and Eric Weisstein's World of Science (ScienceWorld). He is the author of the CRC Concise Encyclopedia of Mathematics. He currently works for Wolfram Research, Inc.

MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana–Champaign.

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[1] Glasser, M. L. "A Remarkable Property of Definite Integrals." Mathematics of Computation 40, 561–563, 1983.

[2] 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

[3] 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