Keyhole contour

A big circle with a thin slit cut along a branch cut and a small circle routed around the branch point, joined into one closed contour.

A keyhole contour is built from four pieces -- a large circle of radius $R$ , a small circle of radius $\varepsilon$ around a branch point, and two nearly-overlapping straight segments hugging a branch cut from either side, joining the two circles into one closed curve with a thin slit left open along the cut. It is the tool for a real integral whose integrand has a branch point (rather than a pole) blocking the plain semicircle or full-circle methods, since neither of those can dodge a cut the integrand is not single-valued across.

Examples

Evaluating $\int_0^\infty x^{a-1}/(1+x)\,dx$ routes a keyhole contour around the branch cut of $z^{a-1}$ along the positive real axis, with the integrand's only pole (at $z=-1$ ) sitting safely inside the slit region.
As $\varepsilon\to0$ and $R\to\infty$ , a keyhole contour's small circle and large circle contribute nothing, leaving only the two straight pieces -- which differ from each other by a fixed factor picked up crossing the branch cut, exactly what makes the original real integral solvable.