- §10.73(i) Bessel and Modified Bessel Functions
- §10.73(ii) Spherical Bessel Functions
- §10.73(iii) Kelvin Functions
- §10.73(iv) Bickley Functions
- §10.73(v) Rayleigh Function

Bessel functions first appear in the investigation of a physical problem in Daniel Bernoulli’s analysis of the small oscillations of a uniform heavy flexible chain. For this problem and its further generalizations, see Korenev (2002, Chapter 4, §37) and Gray et al. (1922, Chapter I, §1, Chapter XVI, §4).

Bessel functions of the first kind, ${J}_{n}\left(x\right)$, arise naturally in applications having cylindrical symmetry in which the physics is described either by Laplace’s equation ${\nabla}^{2}V=0$, or by the Helmholtz equation $({\nabla}^{2}+{k}^{2})\psi =0$.

Laplace’s equation governs problems in heat conduction, in the distribution of potential in an electrostatic field, and in hydrodynamics in the irrotational motion of an incompressible fluid. See Jackson (1999, Chapter 3, §§3.7, 3.8, 3.11, 3.13), Lamb (1932, Chapter V, §§100–102; Chapter VIII, §§186, 191–193; Chapter X, §§303, 304), Happel and Brenner (1973, Chapter 3, §3.3; Chapter 7, §7.3), Korenev (2002, Chapter 4, §43), and Gray et al. (1922, Chapter XI). In cylindrical coordinates $r$, $\varphi $, $z$, (§1.5(ii) we have

10.73.1 | $${\nabla}^{2}V=\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial V}{\partial r}\right)+\frac{1}{{r}^{2}}\frac{{\partial}^{2}V}{{\partial \varphi}^{2}}+\frac{{\partial}^{2}V}{{\partial z}^{2}}=0,$$ | ||

and on separation of variables we obtain solutions of the form ${\mathrm{e}}^{\pm \mathrm{i}n\varphi}{\mathrm{e}}^{\pm \kappa z}{J}_{n}\left(\kappa r\right)$, from which a solution satisfying prescribed boundary conditions may be constructed.

The Helmholtz equation, $({\nabla}^{2}+{k}^{2})\psi =0$, follows from the wave equation

10.73.2 | $${\nabla}^{2}\psi =\frac{1}{{c}^{2}}\frac{{\partial}^{2}\psi}{{\partial t}^{2}},$$ | ||

on assuming a time dependence of the form ${\mathrm{e}}^{\pm \mathrm{i}kt}$. This equation governs problems in acoustic and electromagnetic wave propagation. See Jackson (1999, Chapter 9, §9.6), Jones (1986, Chapters 7, 8), and Lord Rayleigh (1945, Vol. I, Chapter IX, §§200–211, 218, 219, 221a; Vol. II, Chapter XIII, §272a; Chapter XV, §302; Chapter XVIII; Chapter XIX, §350; Chapter XX, §357; Chapter XXI, §369). It is fundamental in the study of electromagnetic wave transmission. Consequently, Bessel functions ${J}_{n}\left(x\right)$, and modified Bessel functions ${I}_{n}\left(x\right)$, are central to the analysis of microwave and optical transmission in waveguides, including coaxial and fiber. See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25).

Bessel functions enter in the study of the scattering of light and other electromagnetic radiation, not only from cylindrical surfaces but also in the statistical analysis involved in scattering from rough surfaces. See Smith (1997, Chapter 3, §3.7; Chapter 6, §6.4), Beckmann and Spizzichino (1963, Chapter 4, §§4.2, 4.3; Chapter 5, §§5.2, 5.3; Chapter 6, §6.1; Chapter 7, §7.1.), Kerker (1969, Chapter 5, §5.6.4; Chapter 7, §7.5.6), and Bayvel and Jones (1981, Chapter 1, §§1.6.5, 1.6.6).

More recently, Bessel functions appear in the inverse problem in wave propagation, with applications in medicine, astronomy, and acoustic imaging. See Colton and Kress (1998, Chapter 2, §§2.4, 2.5; Chapter 3, §3.4). In the theory of plates and shells, the oscillations of a circular plate are determined by the differential equation

10.73.3 | $${\nabla}^{4}W+{\lambda}^{2}\frac{{\partial}^{2}W}{{\partial t}^{2}}=0.$$ | ||

See Korenev (2002). On separation of variables into cylindrical coordinates, the Bessel functions ${J}_{n}\left(x\right)$, and modified Bessel functions ${I}_{n}\left(x\right)$ and ${K}_{n}\left(x\right)$, all appear.

The functions ${\mathsf{j}}_{n}\left(x\right)$, ${\mathsf{y}}_{n}\left(x\right)$, ${\mathsf{h}}_{n}^{(1)}\left(x\right)$, and ${\mathsf{h}}_{n}^{(2)}\left(x\right)$ arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates $\rho ,\theta ,\varphi $ (§1.5(ii)):

10.73.4 | $$({\nabla}^{2}+{k}^{2})f=\frac{1}{{\rho}^{2}}\frac{\partial}{\partial \rho}\left({\rho}^{2}\frac{\partial f}{\partial \rho}\right)+\frac{1}{{\rho}^{2}\mathrm{sin}\theta}\frac{\partial}{\partial \theta}\left(\mathrm{sin}\theta \frac{\partial f}{\partial \theta}\right)+\frac{1}{{\rho}^{2}{\mathrm{sin}}^{2}\theta}\frac{{\partial}^{2}f}{{\partial \varphi}^{2}}+{k}^{2}f.$$ | ||

With the spherical harmonic ${Y}_{\mathrm{\ell},m}\left(\theta ,\varphi \right)$ defined as in §14.30(i), the solutions are of the form $f={g}_{\mathrm{\ell}}(k\rho ){Y}_{\mathrm{\ell},m}\left(\theta ,\varphi \right)$ with ${g}_{\mathrm{\ell}}={\mathsf{j}}_{\mathrm{\ell}}$, ${\mathsf{y}}_{\mathrm{\ell}}$, ${\mathsf{h}}_{\mathrm{\ell}}^{(1)}$, or ${\mathsf{h}}_{\mathrm{\ell}}^{(2)}$, depending on the boundary conditions. Accordingly, the spherical Bessel functions appear in all problems in three dimensions with spherical symmetry involving the scattering of electromagnetic radiation. See Jackson (1999, Chapter 9, §9.6), Bayvel and Jones (1981, Chapter 1, §1.5.1), and Konopinski (1981, Chapter 9, §9.1). In quantum mechanics the spherical Bessel functions arise in the solution of the Schrödinger wave equation for a particle in a central potential. See Messiah (1961, Chapter IX, §§7–10).

The analysis of the current distribution in circular conductors leads to the Kelvin functions $\mathrm{ber}x$, $\mathrm{bei}x$, $\mathrm{ker}x$, and $\mathrm{kei}x$. See Relton (1965, Chapter X, §§10.2, 10.3), Bowman (1958, Chapter III, §§51–53), McLachlan (1961, Chapters VIII and IX), and Russell (1909). The McLachlan reference also includes other applications of Kelvin functions.

For applications of the Rayleigh function ${\sigma}_{n}\left(\nu \right)$ (§10.21(xiii)) to problems of heat conduction and diffusion in liquids see Kapitsa (1951a).