Hankel functions

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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.


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


See Cylinder functions for additional references.

How to Cite This Entry:
Hankel functions. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by P.I. Lizorkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article