Editing Hankel contour (section)

[[File:Hankel Pic.png|thumb|A Hankel contour path, traversed in the positive sense.]]
[[File:Hankel contour.png|thumb|This is a version of the Hankel contour that consists of just a linear mirror image across the real axis.]]
In [[mathematics]], a '''Hankel contour''' is a path in the [[complex plane]] which extends from 
(+∞,δ), around the origin [[counter clockwise]] and back to
(+∞,&minus;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.<ref name=":0">{{Cite book|title=Handbook of complex variables|last=Krantz, Steven G. (Steven George), 1951-|date=1999|publisher=Birkhäuser|isbn=0-8176-4011-8|location=Boston, Mass.|oclc=40964730}}</ref> 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 integral]]s 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 function]]s (which are Bessel functions of the third kind).<ref name=":0" /><ref name=":1">{{Cite book|title=Functions of a Complex Variable|last=Moretti|first=Gino|publisher=Prentice-Hall, Inc.|year=1964|location=Englewood Cliffs, N.J.|pages=179–184|lccn=64012240}}</ref>