Keyhole contour integration method

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

$\int _{0}^{\infty }f(x)\,dx$

There are two motivations for using keyhole contours:

• The function branches. For instance, the logarithm function, or a fractional power map.
• The function has poles on the negative real axis, so we cannot use the semicircular contour integration method or the mousehole contour integration method.

Setting up the complex-valued function

We first choose a meromorphic function $g$ on the plane slit at the positive real axis, such that its limit from one direction, has real or imaginary part equal to $f$

Computing the integral over keyhole contours

The keyhole contour is defined as follows: it has an outer, almost complete circle and an inner, almost complete circle, both centered at the origin. The two circles are joined by straight lines parallel to the real axis. This gives the shape of a keyhole. We take a limit in three ways:

• Send the radius of the outer circle to $\infty$
• Send the radius of the inner circle to $0$
• Send the angle of parting that separates the two ends of the inner circle, to $0$

We then use various limiting techniques to compute the integral on the outer circle, the inner circle, and the two lines.