Glasser's master theorem
# 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 $-\infty$ 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.

## 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

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

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

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