 Hankel contour

A Hankel contour approaches a branch point along one bank of a cut, encircles the point, and returns along the other bank. The orientation and complex logarithm branch determine its jump factors.

For the negative-real cut, let $H$ run from negative infinity below the cut, circle zero counterclockwise, and return above the cut. Use $-\pi <\arg t<\pi$. ThenThis is the Hankel representation of the reciprocal gamma function.^[ref]

When $0<\mathrm{Re}s<1$, shrink the small circle to zero. The lower and upper banks contribute $e^{i\pi s}\Gamma (1-s)$ and $-e^{-i\pi s}\Gamma (1-s)$. Their difference, divided by $2\pi i$, is $\sin (\pi s)\Gamma (1-s)/\pi =1/\Gamma (s)$ by the gamma reflection formula. Keeping a nonzero encircling radius gives continuation beyond that collapse range. Separating divergent bank integrals without this qualification would lose the meaning of the contour representation.

The Hankel contour is a contour in the complex plane commonly used in the context of complex analysis, particularly in the study of integral transforms and asymptotic analysis. It is especially useful for evaluating integrals of functions that have branch cuts or singularities. ### Basic Definition: The Hankel contour typically consists of two parts: 1. A large semicircular arc in the upper half-plane (or lower half-plane depending on the application) that joins two points along the real line.