Bessel equation

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

A second-order linear ordinary differential equation

$$x^2y'' + xy'+(x^2-\nu^2) = 0,\quad \nu = {\rm const},\label{(1)}$$ or, in self-adjoint form:

$$(xy')' + \big(x- \frac{\nu^2}{x}\big)y = 0.$$ 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 (1):

$$u''+\Big(1+\frac{1-4\nu^2}{4x^2}\Big)u = 0.\label{(2)}$$ A Bessel equation is a special case of a confluent hypergeometric equation; if $x=z/2i$ is substituted into (2), equation (2) becomes a Whittaker equation. In equation (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) $J_\nu(x)$, the cylinder functions of the second kind, 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 (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 (1). For this reason, the general solution of equation (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 (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 (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 (1) into the Laplace equation:

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

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

$$x^2y'' + xy' -(\def\l{\lambda}\l x^2-\nu^2) y = 0,\label{(3)}$$ 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 (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

$$x^2y'' + xy'+(x^2-\nu^2)y = f(x)\label{(4)}$$ 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 (4) have been studied in more detail for a right-hand side of special form. Thus, if $f(x) = x^\rho$, equation (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