In mathematics, the Neumann polynomials, introduced by Carl Neumann for the special case , are a sequence of polynomials in used to expand functions in term of Bessel functions.[1]

The first few polynomials are

A general form for the polynomial is

and they have the "generating function"

where J are Bessel functions.

To expand a function f in the form

for , compute

where and c is the distance of the nearest singularity of from .

Examples

An example is the extension

or the more general Sonine formula[2]

where is Gegenbauer's polynomial. Then,

the confluent hypergeometric function

and in particular

the index shift formula

the Taylor expansion (addition formula)

(cf.[3]) and the expansion of the integral of the Bessel function,

are of the same type.

See also

Notes

  1. Abramowitz and Stegun, p. 363, 9.1.82 ff.
  2. Erdélyi et al. 1955 II.7.10.1, p.64
  3. Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. "8.515.1.". In Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. p. 944. ISBN 0-12-384933-0. LCCN 2014010276.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.