Bessel equation
A second-order linear ordinary differential equation
![]() | (1) |
or, in self-adjoint form:
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The number is called the order of the Bessel equation; in the general case
and
assume complex values. The substitution
yields the reduced form of equation (1):
![]() | (2) |
A Bessel equation is a special case of a confluent hypergeometric equation; if is substituted into (2), equation (2) becomes a Whittaker equation. In equation (1) the point
is weakly singular, while the point
is strongly singular. For this reason a Bessel equation does not belong to the class of Fuchsian equations (cf. Fuchsian equation). F. Bessel [1] 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 [2], 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) , the cylinder functions of the second kind (Weber functions or Neumann functions, (cf. Weber function; Neumann function)
and the cylinder functions of the third kind (Hankel functions)
,
. If the order
is fixed, all these functions are analytic functions of the complex argument
; for all these functions, except for the functions
of integer order, the point
is a branch point. If the argument
is fixed, all these functions are single-valued entire functions of the complex order
[3].
If the order is not an integer, then the general solution of equation (1) may be written as
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where are arbitrary constants. For a given order, any two of the functions
,
,
,
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:
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The following equations are closely connected with equation (1): the equation
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which becomes (1) as a result of the substitution , and with as a fundamental system of solutions the modified cylinder functions (Bessel functions of imaginary argument), and the equation
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which becomes equation (1) as a result of the substitution 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 [4].
The substitution transforms (1) into the Laplace equation:
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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 eigen values of the equation
![]() | (3) |
where is fixed while
is a parameter. Equation (3) on the segment
with the boundary conditions:
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is an example of a problem with a discrete spectrum (the eigen values are determined by the condition in terms of the zeros of a Bessel function). Equation (3) with the boundary condition:
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represents a problem with a continuous spectrum (eigen values ).
The inhomogeneous Bessel equation
![]() | (4) |
has the particular solution
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Solutions of equation (4) have been studied in more detail for a right-hand side of special form. Thus, if , equation (4) is satisfied by a Lommel function; if
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it is satisfied by a Struve function; if
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it is satisfied by an Anger function; and if
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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 -th order equation of Bessel type has the form
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and its solution depends on parameters. In particular, a third-order equation of Bessel type (which has a solution with two parameters
,
) may be represented in the form:
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References
[1] | F. Bessel, Abh. d. K. Akad. Wiss. Berlin (1824) pp. 1–52 |
[2] | A. Gray, G.B. Mathews, "A treatise on Bessel functions and their application to physics" , Macmillan (1931) |
[3] | G.N. Watson, "A treatise on the theory of Bessel functions" , 1–2 , Cambridge Univ. Press (1952) |
[4] | E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1971) |
Comments
References
[a1] | N.N. Lebedev, "Special functions and their applications" , Dover, reprint (1972) (Translated from Russian) |
Bessel equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bessel_equation&oldid=31321