Neumann function
From Encyclopedia of Mathematics
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A cylinder function (cf. Cylinder functions) of the second kind. The Neumann functions (occasionally the notation is used) can be defined in terms of the Bessel functions as follows:
They are real for positive real and tend to zero as . For large they have the asymptotic representation
They are connected by the recurrence formulas
For integers :
for small :
where is the Euler constant.
Figure: n066420a
Graphs of Neumann functions.
The Neumann functions of "half-integral" order can be expressed in terms of the trigonometric functions; in particular,
They were introduced by C.G. Neumann in 1867.
For references see Cylinder functions.
How to Cite This Entry:
Neumann function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Neumann_function&oldid=17802
Neumann function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Neumann_function&oldid=17802
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article