In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Hankel contour is a path in the complex plane

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

 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.

Use of Hankel contours is one of the methods of contour integration

Methods of contour integration

In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.Contour integration is closely related to the calculus of residues, a methodology of complex analysis....

. This type of path for contour integrals was first used by Hermann Hankel

Hermann Hankel

Hermann Hankel was a German mathematician who was born in Halle, Germany and died in Schramberg , Imperial Germany....

 in his investigations of the Gamma function

Gamma function

In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...

.

The mirror image

Mirror image

A mirror image is a reflected duplication of an object that appears identical but reversed. As an optical effect it results from reflection off of substances such as a mirror or water. It is also a concept in geometry and can be used as a conceptualization process for 3-D structures...

 extending from −∞, circling the origin clockwise

Clockwise

Circular motion can occur in two possible directions. A clockwise motion is one that proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back to the top...

, and returning
to −∞ is also called a Hankel contour.