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A second-order linear ordinary differential 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>
+
\begin{equation}
 +
x^2y'' + xy'+(x^2-\nu^2)y = 0,\quad \nu = {\rm const},\label{(1)}
 +
\end{equation}
  
 
or, in self-adjoint form:
 
or, in self-adjoint form:
  
<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/b0158302.png" /></td> </tr></table>
+
\begin{equation*}(xy')' + \big(x- \frac{\nu^2}{x}\big)y = 0.\end{equation*}
  
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):
+
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)}:
  
<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>
+
\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|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 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|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|Fuchsian equation]]). F. Bessel
 +
{{Cite|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 [[#References|[2]]], particularly in the case of boundary value problems of potential theory for a cylindrical domain.
+
A Bessel equation results from separation of variables in many problems of mathematical physics
 +
{{Cite|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|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]]].
+
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|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|Neumann function]]) $Y_\nu(x)$ and the cylinder functions of the third kind ([[Hankel functions|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|branch point]]. If the argument $x$ is fixed, all these functions are single-valued entire functions of the complex order $\nu$
 +
{{Cite|Wa}}.
  
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
+
If the order $\nu$ is not an integer, then the general solution of equation \eqref{(1)} may be written as
  
<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/b01583022.png" /></td> </tr></table>
+
$$y=C_1J_\nu(x) + C_2 J_{-\nu}(x),$$
  
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:
 
  
<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/b01583028.png" /></td> </tr></table>
+
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
 
The following equations are closely connected with equation (1): 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/b01583029.png" /></td> </tr></table>
+
$$z^2y'' + zy' -(z^2+\nu^2) y = 0,$$
 
+
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
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
 
 
 
<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/b01583031.png" /></td> </tr></table>
 
 
 
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]]:
 
 
 
<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/b01583034.png" /></td> </tr></table>
 
  
which permits one to represent the solutions of (1) by contour integrals in the complex plane.
+
$$z^2y'' + zy' -(iz^2+\nu^2) y = 0,$$
 +
which becomes equation \eqref{(1)} as a result of the substitution $x\sqrt{ix}$ and which has the
 +
[[Kelvin functions|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
 +
{{Cite|Ka}}.
  
In applications it is often required to find the eigen values of the equation
+
The substitution $y=x^\nu w$ transforms \eqref{(1)} into the
 +
[[Laplace 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>
+
$$xw''+(2\nu+1)w' + xw = 0;$$
 +
which permits one to represent the solutions of \eqref{(1)} by contour integrals in the complex plane.
  
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:
+
In applications it is often required to find the eigenvalues 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/b01583039.png" /></td> </tr></table>
+
\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:
  
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:
+
$$y(x) \text{ is bounded as } x\to 0,\quad y(\a) = 0,$$
 +
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:
  
<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/b01583041.png" /></td> </tr></table>
+
$$y(x) \text{ is bounded on the semi-axis } 0\le x <\infty,$$
 
+
represents a problem with a continuous spectrum (eigenvalues $\l\ge 0$).
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
 
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>
+
\begin{equation}x^2y'' + xy'+(x^2-\nu^2)y = f(x)\label{(4)}\end{equation}
 
 
 
has the particular solution
 
has the particular solution
  
<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/b01583044.png" /></td> </tr></table>
+
$$y = \frac{\pi}{2}Y_\nu(x) \int \frac{J_\nu(x)}{x}f(x)dx - \frac{\pi}{2}J_\nu(x) \int \frac{Y_\nu(x)}{x}f(x)dx$$
 +
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|Lommel function]]; if
  
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
+
$$f(x) = \frac{4(x/2)^{\nu+1}}{\sqrt{\pi}\;\def\G{\Gamma}\G(\nu+1/2)}$$
 +
it is satisfied by a
 +
[[Struve function|Struve function]]; if
  
<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/b01583046.png" /></td> </tr></table>
+
$$f(x) = \frac{1}{\pi}(x-\nu)\sin \nu \pi,$$
 +
it is satisfied by an
 +
[[Anger function|Anger function]]; and if
  
it is satisfied by a [[Struve function|Struve function]]; if
+
$$f(x) = -\frac{1}{\pi}[(x+\nu)+(x-\nu)\cos\nu\pi],$$
 +
it is satisfied by a
 +
[[Weber function|Weber function]].
  
<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/b01583047.png" /></td> </tr></table>
+
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
  
it is satisfied by an [[Anger function|Anger function]]; and if
+
$$\prod_{i=1}^n\Big(x\frac{d}{dx} + c_i\Big)y + x^n y = 0,$$
  
<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/b01583048.png" /></td> </tr></table>
+
$$c_i = {\rm const},\quad \sum_{i=1}^n c_i = 0,$$
 +
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:
  
it is satisfied by a [[Weber function|Weber function]].
+
$$x^3y'''+3x^2y''+[1+9\a\b-3(\a+\b)^2]xy' +[x^3 - 9\a\b(\a+\b)+2(\a+\b)^3]y = 0,$$
 
 
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
 
 
 
<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/b01583051.png" /></td> </tr></table>
 
 
 
<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:
 
 
 
<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/b01583056.png" /></td> </tr></table>
 
 
 
<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/b01583057.png" /></td> </tr></table>
 
 
 
<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/b01583058.png" /></td> </tr></table>
 
  
 +
$$\a,\b = {\rm const}.$$
 
====References====
 
====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>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Be}}||valign="top"| F. Bessel,  ''Abh. d. K. Akad. Wiss. Berlin''  (1824)  pp. 1–52    
 +
|-
 +
|valign="top"|{{Ref|GrMa}}||valign="top"| A. Gray,  G.B. Mathews,  "A treatise on Bessel functions and their application to physics", Macmillan  (1931)     {{MR|0477198}} {{MR|0442319}} 
 +
|-
 +
|valign="top"|{{Ref|Ka}}||valign="top"| E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden", '''1. Gewöhnliche Differentialgleichungen''', Chelsea, reprint (1971) {{MR|MR0466672}} {{ZBL|0354.34001}}
 +
|-
 +
|valign="top"|{{Ref|Le}}||valign="top"| N.N. Lebedev,  "Special functions and their applications", Dover, reprint  (1972)  (Translated from Russian)   {{MR|0350075}}    {{ZBL|0271.33001}}
  
 +
|-
 +
|valign="top"|{{Ref|Wa}}||valign="top"|  G.N. Watson,  "A treatise on the theory of Bessel functions", '''1–2''', Cambridge Univ. Press  (1952)  {{MR|1349110}} {{MR|1570252}} {{MR|0010746}} {{MR|1520278}}    {{ZBL|0849.33001}} {{ZBL|0174.36202}} {{ZBL|0063.08184}}
  
 
+
|-
====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>
 

Latest revision as of 20:00, 22 March 2024

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
How to Cite This Entry:
Bessel equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bessel_equation&oldid=14419
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article