Semicircular contour integration method

Description

The semicircular contour method is a method for computing Cauchy principal values for integrals of real-analytic functions over the whole real line. Specifically, it helps to solve problems of the form:

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(x)dx$

where $f$ is a real-analytic function with no expression for its definite integral.

Setting up the complex-valued function

We first choose a holomorphic function or meromorphic function $g$ such that:

• $g$ is defined on an open subset containing the upper half-plane and the real axis
• The real or imaginary part of the restriction of $g$ to the real axis, is precisely $f$

Computing integrals over semicircular contours

Next, we consider the integral of $g$ over semicircular contours. By a semicircular contour, we mean a closed curve comprising a diameter (along the real axis) and a semicircle (in the upper half-plane) of a circle centered at the origin. We do the following:

• We use the residue theorem to estimate what happens to the integral over the whole semicircular contour, as the radius approaches $\mathrm{\infty }$
• We use other methods to bound the integral along the semicircular part of the contour
• We take the difference and hopefully obtain the Cauchy principal value for the integral along the real axis