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

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