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.

References

External links