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

The master theorem

If,, and are real numbers and
then

Examples

where the first equality comes from cancelling, the second from Cauchy–Schlömilch, and the last one from a substitution and the integral of the arctangent function.