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 $-\infty$ to $+\infty .$ 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.

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

${\text{P.V.}}\int _{-\infty }^{\infty }F\left(x-{\frac {1}{x}}\right)\mathrm {d} x={\text{P.V.}}\int _{-\infty }^{\infty }F(x)\,\mathrm {d} x$

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

The master theorem

If $a$, $a_{n}$, and $b_{n}$ are real numbers and

$\phi (x)=x-a-\sum _{n=1}^{N}{\frac {|a_{n}|}{x-b_{n}}}$

then

$\operatorname {P.V.} \int _{-\infty }^{\infty }F(\phi (x))\,\mathrm {d} x=\operatorname {P.V.} \int _{-\infty }^{\infty }F(x)\,\mathrm {d} x.$

Examples

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

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

External links

• Weisstein, Eric W. "Glasser's Master Theorem". MathWorld.