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]\displaystyle{ -\infty }[/math] to [math]\displaystyle{ +\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.

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

[math]\displaystyle{ u = x - \frac 1 x \, }[/math]

then

[math]\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) }[/math]

where PV denotes the Cauchy principal value.

The master theorem

If [math]\displaystyle{ a }[/math], [math]\displaystyle{ a_i }[/math], and [math]\displaystyle{ b_i }[/math] are real numbers and

[math]\displaystyle{ u = x - a - \sum_{n=1}^N \frac{|a_n|}{x-b_n} }[/math]

then

[math]\displaystyle{ \operatorname{PV} \int_{-\infty}^\infty F(u)\,dx = \operatorname{PV} \int_{-\infty}^\infty F(x)\,dx. }[/math]

Examples

 

• [math]\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}. }[/math]

External links

• Weisstein, Eric W.. "Glasser's Master Theorem". http://mathworld.wolfram.com/GlassersMasterTheorem.html.