Keyhole contour

Definition

The keyhole contour with inner radius $r>0$, outer radius $R>r$, and width $d\in (0,r)$, is defined as the following contour:

• An arc of the circle of radius $R$ centered at the origin, that excludes the part in the right half-plane that has distance less than $d$ from the $x$-axis
• An arc of the circle of radius $r$ centered at the origin, that excludes the part in the right half-plane that has distance less than $d$ from the $x$-axis
• Line segments parallel to the $x$-axis, joining the ends of these arcs.

Keyhole contours are used to compute integrals of real-valued functions. For this we usually send $r,d\to 0,R\to \mathrm{\infty }$.

Further information: Keyhole contour integration method