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Erschienen in:
Introduction and Preliminaries Kapitel lesen Erstes Kapitel lesen

2017 | OriginalPaper | Buchkapitel

2. Neumann Series

verfasst von : Árpád Baricz, Dragana Jankov Maširević, Tibor K. Pogány

Erschienen in: Series of Bessel and Kummer-Type Functions

Verlag: Springer International Publishing

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Abstract

The goal of present chapter is to study in details the integral representations of the Neumann series (of the first and second type) of Bessel and modified Bessel functions of the first and second kind. In order to achieve our goal we use several methods: the Euler–Maclaurin summation technique, differential equation technique, fractional integration technique. Moreover, we present some interesting results on the coefficients of Neumann series, product of modified Bessel functions of the first and second kind and the cumulative distribution function of the non-central χ 2-distribution.

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Vorheriges Kapitel Introduction and Preliminaries
Nächstes Kapitel Kapteyn Series
Fußnoten
1
Watson remarked that all four formulae that were cited by him [333, 16.53 Eqs. (1), (2), (11), (12)] had been derived by von Lommel (cf. von Lommel’s memoirs [324, 325] for further details).
 
2
It is worth to mention here that the above procedure for modified Bessel functions is similar to the method for Bessel functions applied by Wilkins [335]. See also Andrews et al. [7] for more details. More precisely, Wilkins proved that the Hankel functions \(\big (H_{\nu }^{(1)}\big )^2\) and \(\big (H_{\nu }^{(2)}\big )^2,\) as well as \(J_{\nu }^2+Y_{\nu }^2,\) where J ν and Y ν stand for the Bessel functions of the first and second kind, are particular solutions of the third order homogeneous differential equation [7, p. 225]
$$\displaystyle \begin{aligned} x^2y'''(x)+3xy''(x)+(1+4x^2-4\nu^2)y'(x)+4xy(x)=0. \end{aligned}$$
The above result was used to prove the celebrated Nicholson formula [7, p. 224]
$$\displaystyle \begin{aligned}J_{\nu}^2(x)+Y_{\nu}^2(x)=\frac{8}{\pi^2}\int_0^{\infty}K_0(2x\sinh t)\cosh(2\nu t) \mathrm{d}t,\end{aligned}$$
which generalizes the trigonometric identity \(\sin ^2x+\cos ^2x=1.\)
 
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Metadaten
Titel
Neumann Series
verfasst von
Árpád Baricz
Dragana Jankov Maširević
Tibor K. Pogány
Copyright-Jahr
2017
Buch
Series of Bessel and Kummer-Type Functions

Print ISBN: 978-3-319-74349-3

Electronic ISBN: 978-3-319-74350-9

Copyright-Jahr: 2017

DOI
https://doi.org/10.1007/978-3-319-74350-9_2