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Difference between revisions of "Whittaker equation"

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A linear homogeneous ordinary differential equation of the second order:
 
A linear homogeneous ordinary differential equation of the second order:
  
<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/w097840/w0978401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$w''+\left(\frac{1/4-\mu^2}{z^2}+\frac\lambda z-\frac14\right)w=0,\tag{*}$$
  
where the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097840/w0978402.png" /> and the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097840/w0978403.png" /> may take arbitrary complex values. Equation (*) represents the reduced form of a degenerate [[Hypergeometric equation|hypergeometric equation]] and was first studied by E.T. Whittaker [[#References|[1]]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097840/w0978404.png" /> the Whittaker equation is equivalent to the [[Bessel equation|Bessel equation]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097840/w0978405.png" /> is not an integer, a fundamental system of solutions of the Whittaker equation consists of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097840/w0978406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097840/w0978407.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097840/w0978408.png" /> is the Whittaker function (cf. [[Whittaker functions|Whittaker functions]]). For any value of the parameters the general solution of the Whittaker equation may be written in the form of a linear combination
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where the variables $z,w$ and the parameters $\lambda,\mu$ may take arbitrary complex values. Equation \ref{*} represents the reduced form of a degenerate [[Hypergeometric equation|hypergeometric equation]] and was first studied by E.T. Whittaker [[#References|[1]]]. For $\lambda=0$ the Whittaker equation is equivalent to the [[Bessel equation|Bessel equation]]. If $2\mu$ is not an integer, a fundamental system of solutions of the Whittaker equation consists of the functions $M_{\lambda,\mu}(z)$ and $M_{\lambda,-\mu}(z)$; here $M_{\lambda,\mu}(z)$ is the Whittaker function (cf. [[Whittaker functions|Whittaker functions]]). For any value of the parameters the general solution of the Whittaker equation may be written in the form of a linear combination
  
<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/w097840/w0978409.png" /></td> </tr></table>
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$$w=C_1W_{\lambda,\mu}(z)+C_2W_{-\lambda,\mu}(-z),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097840/w09784010.png" /> is the Whittaker function.
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where $W_{\lambda,\mu}(z)$ is the Whittaker function.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.T. Whittaker,  "An expression of certain known functions as generalized hypergeometric functions"  ''Bull. Amer. Math. Soc.'' , '''10'''  (1903)  pp. 125–134</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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''1. The gamma function. The hypergeometric functions. Legendre functions''' , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Kratzer,  W. Franz,  "Transzendente Funktionen" , Akad. Verlagsgesell. Geest u. Portig K.-D.  (1960)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Akad. Verlagsgesell.  (1942)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.T. Whittaker,  "An expression of certain known functions as generalized hypergeometric functions"  ''Bull. Amer. Math. Soc.'' , '''10'''  (1903)  pp. 125–134</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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''1. The gamma function. The hypergeometric functions. Legendre functions''' , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Kratzer,  W. Franz,  "Transzendente Funktionen" , Akad. Verlagsgesell. Geest u. Portig K.-D.  (1960)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Akad. Verlagsgesell.  (1942)</TD></TR></table>

Revision as of 10:59, 1 August 2014

A linear homogeneous ordinary differential equation of the second order:

$$w''+\left(\frac{1/4-\mu^2}{z^2}+\frac\lambda z-\frac14\right)w=0,\tag{*}$$

where the variables $z,w$ and the parameters $\lambda,\mu$ may take arbitrary complex values. Equation \ref{*} represents the reduced form of a degenerate hypergeometric equation and was first studied by E.T. Whittaker [1]. For $\lambda=0$ the Whittaker equation is equivalent to the Bessel equation. If $2\mu$ is not an integer, a fundamental system of solutions of the Whittaker equation consists of the functions $M_{\lambda,\mu}(z)$ and $M_{\lambda,-\mu}(z)$; here $M_{\lambda,\mu}(z)$ is the Whittaker function (cf. Whittaker functions). For any value of the parameters the general solution of the Whittaker equation may be written in the form of a linear combination

$$w=C_1W_{\lambda,\mu}(z)+C_2W_{-\lambda,\mu}(-z),$$

where $W_{\lambda,\mu}(z)$ is the Whittaker function.

References

[1] E.T. Whittaker, "An expression of certain known functions as generalized hypergeometric functions" Bull. Amer. Math. Soc. , 10 (1903) pp. 125–134
[2] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952)
[3] H. Bateman (ed.) A. Erdélyi (ed.) , Higher transcendental functions , 1. The gamma function. The hypergeometric functions. Legendre functions , McGraw-Hill (1953)
[4] A. Kratzer, W. Franz, "Transzendente Funktionen" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1960)
[5] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Akad. Verlagsgesell. (1942)
How to Cite This Entry:
Whittaker equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whittaker_equation&oldid=12547
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article