Mousehole contour integration method: Difference between revisions

Latest revision as of 19:16, 18 May 2008

Description

The mousehole contour integration method is a method used for computing Cauchy principal values for integrals of real-valued functions $f : R^{*} \to R$ , 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:

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

We consider here the function:

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

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 $R_{2} \to \infty$ , and use considerations like Jordan's lemma (and actual computation) to determine the integration along the semicircular parts).

Examples

Computing the sine integral

@@ Line 3: / Line 3: @@
 The '''mousehole contour integration method''' is a method used for computing [[Cauchy principal value]]s for integrals of real-valued functions <math>f: \R^* \to \R</math>, that may blow up at zero.
-===
+===Setting up the complex-valued function===
+We first choose a function <math>g</math>, holomorphic or meromorphic on the upper half-plane as well as on the real line, except possibly at zero, so that <math>f</math> is the real or imaginary part of <math>g</math>.
+For instance, consider the function:
+<math>f(x) = \operatorname{sinc}(x) := \frac{\sin x}{x}</math>
+We consider here the function:
+<math>g(z) = \frac{e^{iz}}{z}</math>
+Although <math>f</math> is real-analytic at <math>0</math>, <math>g</math> has an essential singularity at <math>0</math>.
+If the holomorphic function we construct has a pole of order more than one at the origin, we add to it a polynomial in <math>1/z</math>, so that <math>f</math> continues to remain its imaginary part, but we now have only a simple pole.
+This technique is used to integrate expressions like <math>(x - \sin x)/(x^3)</math>.
+===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 <math>R_1 < R_2</math>, and lines on the real axis joining their ends together. We then make <math>R_1 \to 0</math> and <math>R_2 \to \infty</math>, and use considerations like [[Jordan's lemma]] (and actual computation) to determine the integration along the semicircular parts).
+==Examples==
+* [[Computing the sine integral]]