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:

$f(x)=\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}(x):=\frac{\sin x}{x}$

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 $(x-\sin x)/(x^3)$.

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 $R_2\to \mathrm{\infty }$, and use considerations like Jordan's lemma (and actual computation) to determine the integration along the semicircular parts).

Examples

• Computing the sine integral