Bessel equation

2020 Mathematics Subject Classification: Primary: 34-XX [MSN][ZBL]

A second-order linear ordinary differential equation

\begin{equation} x^2y'' + xy'+(x^2-\nu^2)y = 0,\quad \nu = {\rm const},\label{(1)} \end{equation}

or, in self-adjoint form:

\begin{equation*}(xy')' + \big(x- \frac{\nu^2}{x}\big)y = 0.\end{equation*}

The number $\nu$ is called the order of the Bessel equation; in the general case $x$ and $y$ assume complex values. The substitution $y=ux^{-1/2}$ yields the reduced form of equation \eqref{(1)}:

\begin{equation}u''+\Big(1+\frac{1-4\nu^2}{4x^2}\Big)u = 0.\label{(2)}\end{equation}

A Bessel equation is a special case of a confluent hypergeometric equation; if $x=z/2i$ is substituted into \eqref{(2)}, equation \eqref{(2)} becomes a Whittaker equation. In equation \eqref{(1)} the point $x=0$ is weakly singular, while the point $x=\infty$ is strongly singular. For this reason a Bessel equation does not belong to the class of Fuchsian equations (cf. Fuchsian equation). F. Bessel [Be] was the first to study equation (1) systematically, but such equations are encountered even earlier in the works of D. Bernoulli, L. Euler and J.L. Lagrange.

A Bessel equation results from separation of variables in many problems of mathematical physics [GrMa], particularly in the case of boundary value problems of potential theory for a cylindrical domain.

The solutions of Bessel equations are called cylinder functions (or Bessel functions). These may be subdivided into the cylinder functions of the first kind (Bessel functions of the first kind) $J_\nu(x)$, the cylinder functions of the second kind, commonly called Bessel functions of the second kind, but also Weber functions or Neumann functions, (cf. Neumann function) $Y_\nu(x)$ and the cylinder functions of the third kind (Hankel functions) $H_\nu^{(1)}(x)$, $H_\nu^{(2)}(x)$. If the order $\nu$ is fixed, all these functions are analytic functions of the complex argument $x$; for all these functions, except for the functions $J_\nu(x)$ of integer order, the point $x=0$ is a branch point. If the argument $x$ is fixed, all these functions are single-valued entire functions of the complex order $\nu$ [Wa].

If the order $\nu$ is not an integer, then the general solution of equation \eqref{(1)} may be written as

$$y=C_1J_\nu(x) + C_2 J_{-\nu}(x),$$

where $C_1, C_2$ are arbitrary constants. For a given order, any two of the functions $J_\nu(x)$, $Y_\nu(x)$, $H_\nu^{(1)}(x)$, $H_\nu^{(2)}(x)$ are linearly independent and may serve as a fundamental system of solutions of \eqref{(1)}. For this reason, the general solution of equation \eqref{(1)} can be represented, in particular, in the following forms:

$$y=C_1J_\nu(x) + C_2 Y_\nu(x),\quad y=C_1 H_\nu^{(1)}(x) + C_2 H_\nu^{(2)}(x).$$ The following equations are closely connected with equation (1): the equation

$$z^2y'' + zy' -(z^2+\nu^2) y = 0,$$ which becomes (1) as a result of the substitution $z=ix$, and with as a fundamental system of solutions the modified cylinder functions (Bessel functions of imaginary argument), and the equation

$$z^2y'' + zy' -(iz^2+\nu^2) y = 0,$$ which becomes equation \eqref{(1)} as a result of the substitution $x\sqrt{ix}$ and which has the Kelvin functions as its fundamental system of solutions. Many other second-order linear ordinary differential equations (e.g. the Airy equation) can also be transformed into equation \eqref{(1)} by a transformation of the unknown function and the independent variable. The solution of a series of linear equations of higher orders may be written in the form of Bessel functions [Ka].

The substitution $y=x^\nu w$ transforms \eqref{(1)} into the Laplace equation:

$$xw''+(2\nu+1)w' + xw = 0;$$ which permits one to represent the solutions of \eqref{(1)} by contour integrals in the complex plane.

In applications it is often required to find the eigenvalues of the equation

\begin{equation}x^2y'' + xy' -(\def\l{\lambda}\l x^2-\nu^2) y = 0,\label{(3)}\end{equation} where $\nu$ is fixed while $\l$ is a parameter. Equation (3) on the segment $0\le x\le \def\a{\alpha}\a$ with the boundary conditions:

$$y(x) \text{ is bounded as } x\to 0,\quad y(\a) = 0,$$ is an example of a problem with a discrete spectrum (the eigenvalues are determined by the condition $J_\nu(\a\sqrt{\l})=0$ in terms of the zeros of a Bessel function). Equation \eqref{(3)} with the boundary condition:

$$y(x) \text{ is bounded on the semi-axis } 0\le x <\infty,$$ represents a problem with a continuous spectrum (eigenvalues $\l\ge 0$).

The inhomogeneous Bessel equation

\begin{equation}x^2y'' + xy'+(x^2-\nu^2)y = f(x)\label{(4)}\end{equation} has the particular solution

$$y = \frac{\pi}{2}Y_\nu(x) \int \frac{J_\nu(x)}{x}f(x)dx - \frac{\pi}{2}J_\nu(x) \int \frac{Y_\nu(x)}{x}f(x)dx$$ Solutions of equation \eqref{(4)} have been studied in more detail for a right-hand side of special form. Thus, if $f(x) = x^\rho$, equation \eqref{(4)} is satisfied by a Lommel function; if

$$f(x) = \frac{4(x/2)^{\nu+1}}{\sqrt{\pi}\;\def\G{\Gamma}\G(\nu+1/2)}$$ it is satisfied by a Struve function; if

$$f(x) = \frac{1}{\pi}(x-\nu)\sin \nu \pi,$$ it is satisfied by an Anger function; and if

$$f(x) = -\frac{1}{\pi}[(x+\nu)+(x-\nu)\cos\nu\pi],$$ it is satisfied by a Weber function.

There are linear equations of higher orders with solutions whose properties are analogous to those of Bessel functions. The general $n$-th order equation of Bessel type has the form

$$\prod_{i=1}^n\Big(x\frac{d}{dx} + c_i\Big)y + x^n y = 0,$$

$$c_i = {\rm const},\quad \sum_{i=1}^n c_i = 0,$$ and its solution depends on $n-1$ parameters. In particular, a third-order equation of Bessel type (which has a solution with two parameters $\a$, $\def\b{\beta}\b$) may be represented in the form:

$$x^3y'''+3x^2y''+[1+9\a\b-3(\a+\b)^2]xy' +[x^3 - 9\a\b(\a+\b)+2(\a+\b)^3]y = 0,$$

$$\a,\b = {\rm const}.$$

References

[Be]	F. Bessel, Abh. d. K. Akad. Wiss. Berlin (1824) pp. 1–52
[GrMa]	A. Gray, G.B. Mathews, "A treatise on Bessel functions and their application to physics", Macmillan (1931) MR0477198 MR0442319
[Ka]	E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden", 1. Gewöhnliche Differentialgleichungen, Chelsea, reprint (1971) MRMR0466672 Zbl 0354.34001
[Le]	N.N. Lebedev, "Special functions and their applications", Dover, reprint (1972) (Translated from Russian) MR0350075 Zbl 0271.33001
[Wa]	G.N. Watson, "A treatise on the theory of Bessel functions", 1–2, Cambridge Univ. Press (1952) MR1349110 MR1570252 MR0010746 MR1520278 Zbl 0849.33001 Zbl 0174.36202 Zbl 0063.08184