Bessel Functions in \(\mathbb{R}^q\)
By virtue of polar coordinates, metaharmonic functions, i.e., the solutions of the Helmholtz equation (Δ+λ)U=0, can be decomposed into a radial and an angular part. The Funk–Hecke formula (see Theorem 6.5.5) serves as the appropriate tool for decompositio
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Bessel Functions in Rq
By virtue of polar coordinates, metaharmonic functions, i.e., the solutions of the Helmholtz equation . C /U D 0, can be decomposed into a radial and an angular part. The Funk–Hecke formula (see Theorem 6.5.5) serves as the appropriate tool for decomposition. The angular part leads back to spherical harmonics, while the radial part satisfies a characteristic differential equation. Its solutions are the Bessel functions. This chapter deals with the Bessel functions of dimension q. Dependent on the choice of the parameter 2 Rnf0g we obtain different types of Bessel functions such as regular Bessel functions (in Sect. 8.1), modified Bessel functions (in Sect. 8.2), Hankel functions (in Sect. 8.3), Neumann functions as well as Kelvin functions (in Sect. 8.4). The approach can be regarded as a multi-variate extension of Chap. 7. It closely follows the concept proposed by M¨uller (1998) by which a large number of results known for q D 2; 3 (e.g., occurring in the survey article Niemeyer (1962)) can be generalized to q-dimensional nomenclature in an appropriate way. The exercises in Sect. 8.6 relate to boundary-value problems for the Helmholtz equation corresponding to spherical boundary values. The principle for solution is the superposition of metaharmonic functions consisting of certain products of Bessel functions and spherical harmonics leading to convergent series expansions in the inside and/or the outside of a ball. Particular focus is on the behavior at infinity of entire solutions.
8.1 Regular Bessel Functions By a simple coordinate transformation the equation . C / U D 0; 2 R n f0g, can be reduced to . C 1/U D 0 or . 1/U D 0, respectively. The best-known solutions of these Helmholtz equations are x 7! eix or x 7! ex ; x 2 Rq ; 2 Sq1 , respectively. Since the Helmholtz operators ˙ 1 are linear, more general solutions can be obtained by superposition. This is the basic idea of (regular) Bessel W. Freeden and M. Gutting, Special Functions of Mathematical (Geo-)Physics, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-0348-0563-6 8, © Springer Basel 2013
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8 Bessel Functions in Rq
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and Hankel functions (for more details the reader is referred to, e.g., M¨uller 1952, 1998; Sommerfeld 1966; Watson 1944 and the references therein). The point of departure for our work is the entire solution Un W Rq ! C of the Helmholtz equation Un C Un D 0 in Rq of the form Un .x/ D
in kSq1 k
Z Sq1
eix Yn .qI / dS./;
(8.1.1)
where Yn .qI / is a member of Harmn .Sq1 /. In terms of (standard) polar coordinates x D r; r D jxj, 2 Sq1 , the Funk–Hecke formula (see Theorem 6.5.5) yields the decomposition Un .r/ D in
kSq2 k kSq1 k
Z
1 1
eirt Pn .qI t/.1 t 2 /
q3 2
dt Yn .qI /:
(8.1.2)
In other words, a separation of the variables into a radial and an angular part is achieved in the form Un .x/ D Jn .qI r/ Yn .qI /;
x D r;
2 Sq1 :
(8.1.3)
Definition 8.1.1. The function Jn .qI / given by Jn .qI r/ D in
kSq2 k kSq1 k
Z
1 1
eirt
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