Hankel contour

In mathematics, a Hankel contour is a path in the complex plane which extends from (+∞,δ), around the origin counter clockwise and back to (+∞,−δ), where δ is an arbitrarily small positive number. The contour thus remains arbitrarily close to the real axis but without crossing the real axis except for negative values of x. The Hankel contour can also be represented by a path that has mirror images just above and below the real axis, connected to a circle of radius ε, centered at the origin, where ε is an arbitrarily small number. The two linear portions of the contour are said to be a distance of δ from the real axis. Thus, the total distance between the linear portions of the contour is 2δ.^[1] The contour is traversed in the positively-oriented sense, meaning that the circle around the origin is traversed counter-clockwise.

The general principle is that δ and ε are infinitely small and that the integration contour does not envelop any non-analytic point of the function to be integrated except possibly, in zero. Under these conditions, in accordance with Cauchy's theorem, the value of the integral is the same regardless of δ and ε. Usually, the operation consists of calculating first the integral for non zero values of δ and ε, and then making them tend to 0.

Use of Hankel contours is one of the methods of contour integration. This type of path for contour integrals was first explicitly used by Hermann Hankel in his investigations of the Gamma function, though Riemann already implicitly used it in his paper on the Riemann zeta function in 1859.

The Hankel contour is used to evaluate integrals such as the Gamma function, the Riemann zeta function, and other Hankel functions (which are Bessel functions of the third kind).^[1][2]