Neumann polynomial

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In mathematics, a Neumann polynomial, introduced by Carl Neumann for the special case , is a polynomial in 1/z used to expand functions in term of Bessel functions.[1]

The first few polynomials are

A general form for the polynomial is

they have the generating function

where J are Bessel functions.

To expand a function f in form

for compute

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

Examples[edit]

An example is the extension

or the more general Sonine formula[2]

where is Gegenbauer's polynomial. Then,[citation needed][original research?]

the confluent hypergeometric function

and in particular

the index shift formula

the Taylor expansion (addition formula)

(cf.[3][not in citation given]) and the expansion of the integral of the Bessel function

are of the same type.

See also[edit]

Notes[edit]

  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. 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. ISBN 978-0-12-384933-5.