A Hankel contour runs along both banks of a branch cut and circles its branch point, converting the jump of a complex power into a sine factor.
Subtracting the local Taylor expansion at the encircled branch point extends a Hankel integral successively across left half-planes; prefactors often remove the introduced poles.
When the complex power becomes an integer power, a collapsed Hankel contour extracts the coefficient of in the local Laurent series.
Keep the circular portion of a Hankel contour at fixed nonzero radius. Its numerator is an entire function of , because the contour avoids zero and its tails decay exponentially, uniformly on compact parameter sets. At a positive integer , the integrand is single-valued and entire in , so the bank integrals cancel and the circular integral is zero. Thus , canceling the simple zero of in the Hankel representation of the gamma function. At a nonpositive integer , the circular integral instead gives , yielding residue .
Articles by others on the same topic
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.