# Hankel functions

From Encyclopedia of Mathematics

Cylinder functions of the third kind. They may be defined in terms of Bessel functions as follows:

where is not an integer. This implies the important relations

Hankel functions are complex for real values of ; however,

are real if is real and positive. Hankel functions have simple asymptotic representations for large :

The Hankel function of a "half-integral" argument can be expressed in terms of elementary functions, in particular:

These functions were introduced by H. Hankel in 1869.

#### References

[1] | E. Jahnke, F. Emde, F. Lösch, "Tafeln höheren Funktionen" , Teubner (1966) |

#### Comments

See Cylinder functions for additional references.

**How to Cite This Entry:**

Hankel functions.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Hankel_functions&oldid=17400

This article was adapted from an original article by P.I. Lizorkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article