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Difference between revisions of "Weber equation"

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A second-order ordinary linear differential equation:
 
A second-order ordinary linear 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/w/w097/w097310/w0973101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$y''+\left(\nu+\frac12-\frac{x^2}{4}\right)y=0,\quad\nu=\text{const},\tag{*}$$
  
in which the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097310/w0973102.png" /> is strongly singular (cf. [[Singular point|Singular point]]). An equation of this type was first studied by H. Weber in potential theory in connection with the parabolic cylinder [[#References|[1]]]; it is the result of separation of variables for the [[Laplace equation|Laplace equation]] in parabolic coordinates. The substitution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097310/w0973103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097310/w0973104.png" /> converts the Weber equation to the [[Whittaker equation|Whittaker equation]]. It is a special case of a [[Confluent hypergeometric equation|confluent hypergeometric equation]]. The substitution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097310/w0973105.png" /> converts Weber's equation into
+
in which the point $x=\infty$ is strongly singular (cf. [[Singular point|Singular point]]). An equation of this type was first studied by H. Weber in potential theory in connection with the parabolic cylinder [[#References|[1]]]; it is the result of separation of variables for the [[Laplace equation|Laplace equation]] in parabolic coordinates. The substitution $y=x^{-1/2}w$, $z=x^2/2$ converts the Weber equation to the [[Whittaker equation|Whittaker equation]]. It is a special case of a [[Confluent hypergeometric equation|confluent hypergeometric equation]]. The substitution $y=u\exp(-x^2/4)$ converts Weber's equation into
  
<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/w/w097/w097310/w0973106.png" /></td> </tr></table>
+
$$u''-xu'+\nu u=0.$$
  
Solutions of equation (*) are known as parabolic cylinder functions or as Weber–Hermite functions. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097310/w0973107.png" /> is a non-negative integer, equation (*) is satisfied by the function
+
Solutions of equation \ref{*} are known as parabolic cylinder functions or as Weber–Hermite functions. In particular, if $\nu$ is a non-negative integer, equation \ref{*} is satisfied by the 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/w/w097/w097310/w0973108.png" /></td> </tr></table>
+
$$y=\exp(-x^2/4)H_\nu(x),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097310/w0973109.png" /> is the Hermite polynomial (cf. [[Hermite polynomials|Hermite polynomials]]) [[#References|[2]]], [[#References|[3]]], [[#References|[4]]].
+
where $H_\nu(x)$ is the Hermite polynomial (cf. [[Hermite polynomials|Hermite polynomials]]) [[#References|[2]]], [[#References|[3]]], [[#References|[4]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H.F. Weber,  "Ueber die Integration der partiellen Differentialgleichung <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097310/w09731010.png" />"  ''Math. Ann.'' , '''1'''  (1869)  pp. 1–36</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , Cambridge Univ. Press  (1952)  pp. Chapt. 2</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.)  et al. (ed.) , ''Higher transcendental functions'' , '''2. Bessel functions, parabolic cylinder functions, orthogonal polynomials''' , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E. Jahnke,  F. Emde,  "Tables of functions with formulae and curves" , Dover, reprint  (1945)  (Translated from German)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H.F. Weber,  "Ueber die Integration der partiellen Differentialgleichung <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097310/w09731010.png" />"  ''Math. Ann.'' , '''1'''  (1869)  pp. 1–36</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , Cambridge Univ. Press  (1952)  pp. Chapt. 2</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.)  et al. (ed.) , ''Higher transcendental functions'' , '''2. Bessel functions, parabolic cylinder functions, orthogonal polynomials''' , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E. Jahnke,  F. Emde,  "Tables of functions with formulae and curves" , Dover, reprint  (1945)  (Translated from German)</TD></TR></table>

Revision as of 08:15, 8 August 2014

A second-order ordinary linear differential equation:

$$y''+\left(\nu+\frac12-\frac{x^2}{4}\right)y=0,\quad\nu=\text{const},\tag{*}$$

in which the point $x=\infty$ is strongly singular (cf. Singular point). An equation of this type was first studied by H. Weber in potential theory in connection with the parabolic cylinder [1]; it is the result of separation of variables for the Laplace equation in parabolic coordinates. The substitution $y=x^{-1/2}w$, $z=x^2/2$ converts the Weber equation to the Whittaker equation. It is a special case of a confluent hypergeometric equation. The substitution $y=u\exp(-x^2/4)$ converts Weber's equation into

$$u''-xu'+\nu u=0.$$

Solutions of equation \ref{*} are known as parabolic cylinder functions or as Weber–Hermite functions. In particular, if $\nu$ is a non-negative integer, equation \ref{*} is satisfied by the function

$$y=\exp(-x^2/4)H_\nu(x),$$

where $H_\nu(x)$ is the Hermite polynomial (cf. Hermite polynomials) [2], [3], [4].

References

[1] H.F. Weber, "Ueber die Integration der partiellen Differentialgleichung " Math. Ann. , 1 (1869) pp. 1–36
[2] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 2
[3] H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953)
[4] E. Jahnke, F. Emde, "Tables of functions with formulae and curves" , Dover, reprint (1945) (Translated from German)
How to Cite This Entry:
Weber equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weber_equation&oldid=13645
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article