Английская Википедия:Glasser's master theorem
In integral calculus, Glasser's master theorem explains how a certain broad class of substitutions can simplify certain integrals over the whole interval from <math>-\infty</math> to <math>+\infty.</math> 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 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
- <math> u = x - \frac 1 x \, </math>
then
- <math> \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) </math>
where PV denotes the Cauchy principal value.
The master theorem
If <math>a</math>, <math>a_i</math>, and <math>b_i</math> are real numbers and
- <math> u = x - a - \sum_{n=1}^N \frac{|a_n|}{x-b_n} </math>
then
- <math> \operatorname{PV} \int_{-\infty}^\infty F(u)\,dx = \operatorname{PV} \int_{-\infty}^\infty F(x)\,dx. </math>
Examples
- <math> \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}. </math>
References
External links
- ↑ 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