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 real line from ${\displaystyle -\infty }$ to ${\displaystyle +\infty .}$ The integrals in question 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

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

then

${\displaystyle \operatorname {PV} \int _{-\infty }^{\infty }F(u)\,dx=\operatorname {PV} \int _{-\infty }^{\infty }F(x)\,dx\qquad ({\text{Note: }}F(u)\,dx,{\text{ not }}F(u)\,du)}$

where PV denotes the Cauchy principal value and ${\displaystyle F}$ is a function which is integrable on the real line at least in the sense of the Cauchy principal value.

The master theorem

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

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

then

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

Examples

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

where the first equality comes from cancelling ${\displaystyle x^{2}}$, the second from Cauchy–Schlömilch, and the last one from a substitution and the integral of the arctangent function.