Neumann function

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.