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Difference between revisions of "Hermite equation"

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A linear homogeneous second-order ordinary differential equation
 
A linear homogeneous second-order 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/h/h046/h046970/h0469701.png" /></td> </tr></table>
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$$w''-2zw'+\lambda w=0$$
  
 
or, in self-adjoint form,
 
or, in self-adjoint form,
  
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$$\frac{d}{dz}\left(e^{-z^2}\frac{dw}{dz}\right)+\lambda e^{-z^2}w=0;$$
  
here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046970/h0469703.png" /> is a constant. The change of the unknown function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046970/h0469704.png" /> transforms the Hermite equation into
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here $\lambda$ is a constant. The change of the unknown function $w=u\exp(z^2/2)$ transforms the Hermite equation into
  
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$$u''+(\lambda+1-z^2)u=0$$
  
 
and after the change of variables
 
and after the change of variables
  
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$$w=v\exp(t^2/4),\quad t=z\sqrt2$$
  
 
one obtains from the Hermite equation the [[Weber equation|Weber equation]]
 
one obtains from the Hermite equation the [[Weber equation|Weber equation]]
  
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$$v''+\left(\frac\lambda2+\frac12-\frac{t^2}{4}\right)v=0.$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046970/h0469708.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046970/h0469709.png" /> is a natural number, the Hermite equation has among its solutions the Hermite polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046970/h04697010.png" /> (cf. [[Hermite polynomials|Hermite polynomials]]),
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For $\lambda=2n$, where $n$ is a natural number, the Hermite equation has among its solutions the Hermite polynomial of degree $n$ (cf. [[Hermite polynomials|Hermite polynomials]]),
  
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$$H_n(z)=(-1)^ne^{z^2}\frac{d^n}{dz^n}(e^{-z^2}).$$
  
 
This explains the name of the differential equation. In general, the solutions of the Hermite equation can be expressed in terms of special functions: the parabolic cylinder functions or Weber–Hermite functions.
 
This explains the name of the differential equation. In general, the solutions of the Hermite equation can be expressed in terms of special functions: the parabolic cylinder functions or Weber–Hermite functions.

Revision as of 11:52, 3 August 2014

A linear homogeneous second-order ordinary differential equation

$$w''-2zw'+\lambda w=0$$

or, in self-adjoint form,

$$\frac{d}{dz}\left(e^{-z^2}\frac{dw}{dz}\right)+\lambda e^{-z^2}w=0;$$

here $\lambda$ is a constant. The change of the unknown function $w=u\exp(z^2/2)$ transforms the Hermite equation into

$$u''+(\lambda+1-z^2)u=0$$

and after the change of variables

$$w=v\exp(t^2/4),\quad t=z\sqrt2$$

one obtains from the Hermite equation the Weber equation

$$v''+\left(\frac\lambda2+\frac12-\frac{t^2}{4}\right)v=0.$$

For $\lambda=2n$, where $n$ is a natural number, the Hermite equation has among its solutions the Hermite polynomial of degree $n$ (cf. Hermite polynomials),

$$H_n(z)=(-1)^ne^{z^2}\frac{d^n}{dz^n}(e^{-z^2}).$$

This explains the name of the differential equation. In general, the solutions of the Hermite equation can be expressed in terms of special functions: the parabolic cylinder functions or Weber–Hermite functions.


Comments

References

[a1] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1965)
How to Cite This Entry:
Hermite equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermite_equation&oldid=32694
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article