View source for Bessel equation

A second-order linear ordinary differential equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b0158301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

or, in self-adjoint form:

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The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b0158303.png" /> is called the order of the Bessel equation; in the general case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b0158304.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b0158305.png" /> assume complex values. The substitution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b0158306.png" /> yields the reduced form of equation (1):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b0158307.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

A Bessel equation is a special case of a [[Confluent hypergeometric equation|confluent hypergeometric equation]]; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b0158308.png" /> is substituted into (2), equation (2) becomes a [[Whittaker equation|Whittaker equation]]. In equation (1) the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b0158309.png" /> is weakly singular, while the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583010.png" /> is strongly singular. For this reason a Bessel equation does not belong to the class of Fuchsian equations (cf. [[Fuchsian equation|Fuchsian equation]]). F. Bessel [[#References|[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 [[#References|[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|cylinder functions]] (or Bessel functions). These may be subdivided into the cylinder functions of the first kind ([[Bessel functions|Bessel functions]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583011.png" />, the cylinder functions of the second kind (Weber functions or Neumann functions, (cf. [[Weber function|Weber function]]; [[Neumann function|Neumann function]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583012.png" /> and the cylinder functions of the third kind ([[Hankel functions|Hankel functions]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583014.png" />. If the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583015.png" /> is fixed, all these functions are analytic functions of the complex argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583016.png" />; for all these functions, except for the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583017.png" /> of integer order, the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583018.png" /> is a [[Branch point|branch point]]. If the argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583019.png" /> is fixed, all these functions are single-valued entire functions of the complex order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583020.png" /> [[#References|[3]]].

If the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583021.png" /> is not an integer, then the general solution of equation (1) may be written as

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583023.png" /> are arbitrary constants. For a given order, any two of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583027.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583030.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583032.png" /> and which has the [[Kelvin functions|Kelvin functions]] as its fundamental system of solutions. Many other second-order linear ordinary differential equations (e.g. the [[Airy equation|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 [[#References|[4]]].

The substitution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583033.png" /> transforms (1) into the [[Laplace equation|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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583036.png" /> is fixed while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583037.png" /> is a parameter. Equation (3) on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583038.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583040.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583042.png" />).

The inhomogeneous Bessel equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583043.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583045.png" />, equation (4) is satisfied by a [[Lommel function|Lommel function]]; if

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it is satisfied by a [[Struve function|Struve function]]; if

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it is satisfied by an [[Anger function|Anger function]]; and if

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it is satisfied by a [[Weber function|Weber function]].

There are linear equations of higher orders with solutions whose properties are analogous to those of Bessel functions. The general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583050.png" />-th order equation of Bessel type has the form

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583052.png" /></td> </tr></table>

and its solution depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583053.png" /> parameters. In particular, a third-order equation of Bessel type (which has a solution with two parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015830/b01583055.png" />) may be represented in the form:

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====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Bessel,   ''Abh. d. K. Akad. Wiss. Berlin''  (1824)  pp. 1–52</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Gray,   G.B. Mathews,   "A treatise on Bessel functions and their application to physics" , Macmillan  (1931)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.N. Watson,   "A treatise on the theory of Bessel functions" , '''1–2''' , Cambridge Univ. Press  (1952)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E. Kamke,   "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1971)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.N. Lebedev,   "Special functions and their applications" , Dover, reprint  (1972)  (Translated from Russian)</TD></TR></table>