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.
Use of Hankel contours is one of the methods of contour integration. This type of path for contour integrals was first used by Hermann Hankel in his investigations of the Gamma function.
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]
- ^ a b Krantz, Steven G. (Steven George), 1951- (1999). Handbook of complex variables. Boston, Mass.: Birkhäuser. ISBN 0-8176-4011-8. OCLC 40964730.
{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link) - ^ Moretti, Gino (1964). Functions of a Complex Variable. Englewood Cliffs, N.J.: Prentice-Hall, Inc. pp. 179–184. LCCN 64012240.