Mousehole contour integration method

Description

The mousehole contour integration method is a method used for computing Cauchy principal values for integrals of real-valued functions $f:\mathbb {R} ^{*}\to \mathbb {R}$, that may blow up at zero.

Setting up the complex-valued function

We first choose a function $g$, holomorphic or meromorphic on the upper half-plane as well as on the real line, except possibly at zero, so that $f$ is the real or imaginary part of $g$.

For instance, consider the function:

$f(x)=\operatorname {sinc} (x):={\frac {\sin x}{x}}$

We consider here the function:

$g(z)={\frac {e^{iz}}{z}}$

Although $f$ is real-analytic at $0$, $g$ has an essential singularity at $0$.

If the holomorphic function we construct has a pole of order more than one at the origin, we add to it a polynomial in $1/z$, so that $f$ continues to remain its imaginary part, but we now have only a simple pole.

This technique is used to integrate expressions like $(\sin ^{2}x)/(x^{2})$ and $(x-\sin x)/(x^{3})$.

Computing integrals over mousehole contours

A mousehole contour is described as follows: it is made up of semicircles in the upper half-plane of radii $R_{1}<R_{2}$, and lines on the real axis joining their ends together. We then make $R_{1}\to 0$ and $R_{2}\to \infty$, and use considerations like Jordan's lemma (and actual computation) to determine the integration along the semicircular parts).

Examples

• Computing the sine integral