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Glasser's master theorem

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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.

A special case: the Cauchy–Schlömilch transformation

A special case called the Cauchy–Schlömilch substitution or Cauchy–Schlömilch transformation was known to Cauchy in the early 19th century. It states that if

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

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)}

where PV denotes the Cauchy principal value.

The master theorem

If a {\displaystyle a} , a i {\displaystyle a_{i}} , and b i {\displaystyle b_{i}} 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}}}}

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.}

Examples

 

  • ∫ − ∞ ∞ x 2 d x x 4 + 1 = ∫ − ∞ ∞ d x ( x − 1 x ) 2 + 2 = ∫ − ∞ ∞ d x x 2 + 2 = π 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}}}.}

References

  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

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