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A linear ordinary second-order differential equation in the complex domain
 
A linear ordinary second-order differential equation in the complex domain
  
<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/l/l057/l057400/l0574001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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\begin{equation} \label{eq1}
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\frac{d^2 w}{d z^2} = \left [ A + B {\wp} ( z) \right ] w ,
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\end{equation}
 +
 
 +
where  $  {\wp} (z) $
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is the [[Weierstrass p-function|Weierstrass $\wp$-function]] and $A$ and $B$
 +
are constants. This equation was first studied by G. Lamé [[#References|[1]]]; it arises in the separation of variables for the Laplace equation in elliptic coordinates. Equation \eqref{eq1} is also called the Weierstrass form of the Lamé equation. By a change of the independent variable in \eqref{eq1} one obtains Jacobi's form of the Lamé equation:
 +
 
 +
\begin{equation*}
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\frac{d  ^ {2} w }{d u^2} = \left [ C + D  \mathop{\rm sn}  ^ {2}  u \right ] w .
 +
\end{equation*}
 +
 
 +
There are also numerous algebraic forms of the Lamé equation, transition to which is carried out by various transformations of the independent variable in \eqref{eq1}, for example:
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\begin{equation} \label{eq2}
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\frac{d  ^ {2} w }{d \xi  ^ {2} }
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+
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\frac{1}{2}
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\left (
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\frac{1}{\xi - e _ {1} }
 +
+
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057400/l0574002.png" /> is the [[Weierstrass p-function|Weierstrass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057400/l0574003.png" />-function]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057400/l0574004.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057400/l0574005.png" /> are constants. This equation was first studied by G. Lamé [[#References|[1]]]; it arises in the separation of variables for the Laplace equation in elliptic coordinates. Equation (1) is also called the Weierstrass form of the Lamé equation. By a change of the independent variable in (1) one obtains Jacobi's form of the Lamé equation:
+
\frac{1}{\xi - e _ {2} }
 +
+
  
<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/l/l057/l057400/l0574006.png" /></td> </tr></table>
+
\frac{1}{\xi - e _ {3} }
  
There are also numerous algebraic forms of the Lamé equation, transition to which is carried out by various transformations of the independent variable in (1), for example:
+
\right )
  
<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/l/l057/l057400/l0574007.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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\frac{dw}{d \xi }
 +
=
  
<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/l/l057/l057400/l0574008.png" /></td> </tr></table>
+
\frac{A + B \xi }{4 ( \xi - e _ {1} ) ( \xi - e _ {2} ) ( \xi - e _ {3} ) }
 +
w .
 +
\end{equation}
  
For practical applications the Jacobi form is the most suitable.
+
For practical applications, the Jacobi form is the most suitable.
  
Especially important is the case when in (1) (or (2)) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057400/l0574009.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057400/l05740010.png" /> is a natural number. In this case the solutions of (1) are meromorphic in the whole plane and their properties have been thoroughly studied. Among the solutions of (2) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057400/l05740011.png" /> the Lamé functions are of paramount importance (cf. [[Lamé function|Lamé function]]).
+
Especially important is the case when in \eqref{eq1} (or in \eqref{eq2}) $B = n ( n + 1 )$,  
 +
where $n$ is a natural number. In this case the solutions of \eqref{eq1} are meromorphic in the whole plane and their properties have been thoroughly studied. Among the solutions of \eqref{eq2} with  $B = n ( n + 1 )$
 +
the Lamé functions are of paramount importance (cf. [[Lamé function]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Lamé,   "Sur les surfaces isothermes dans les corps homogènes en équilibre de température"  ''J. Math. Pures Appl.'' , '''2'''  (1837)  pp. 147–188</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.J.O. Strutt,  "Lamésche, Mathieusche und Verwandte Funktionen in Physik und Technik"  ''Ergebn. Math.'' , '''1''' :  3  (1932)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , Cambridge Univ. Press  (1952)  pp. Chapt. 6</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''3. Automorphic functions''' , McGraw-Hill  (1955)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.W. Hobson,  "The theory of spherical and ellipsoidal harmonics" , Cambridge Univ. Press  (1931)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  G. Lamé, "Sur les surfaces isothermes dans les corps homogènes en équilibre de température"  ''J. Math. Pures Appl.'' , '''2'''  (1837)  pp. 147–188</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  M.J.O. Strutt,  "Lamésche, Mathieusche und Verwandte Funktionen in Physik und Technik"  ''Ergebn. Math.'' , '''1''' :  3  (1932)</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , Cambridge Univ. Press  (1952)  pp. Chapt. 6</TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''3. Automorphic functions''' , McGraw-Hill  (1955)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.W. Hobson,  "The theory of spherical and ellipsoidal harmonics" , Cambridge Univ. Press  (1931)</TD></TR>
 +
</table>

Latest revision as of 18:25, 1 May 2024


A linear ordinary second-order differential equation in the complex domain

\begin{equation} \label{eq1} \frac{d^2 w}{d z^2} = \left [ A + B {\wp} ( z) \right ] w , \end{equation}

where $ {\wp} (z) $ is the Weierstrass $\wp$-function and $A$ and $B$ are constants. This equation was first studied by G. Lamé [1]; it arises in the separation of variables for the Laplace equation in elliptic coordinates. Equation \eqref{eq1} is also called the Weierstrass form of the Lamé equation. By a change of the independent variable in \eqref{eq1} one obtains Jacobi's form of the Lamé equation:

\begin{equation*} \frac{d ^ {2} w }{d u^2} = \left [ C + D \mathop{\rm sn} ^ {2} u \right ] w . \end{equation*}

There are also numerous algebraic forms of the Lamé equation, transition to which is carried out by various transformations of the independent variable in \eqref{eq1}, for example:

\begin{equation} \label{eq2} \frac{d ^ {2} w }{d \xi ^ {2} } + \frac{1}{2} \left ( \frac{1}{\xi - e _ {1} } + \frac{1}{\xi - e _ {2} } + \frac{1}{\xi - e _ {3} } \right ) \frac{dw}{d \xi } = \frac{A + B \xi }{4 ( \xi - e _ {1} ) ( \xi - e _ {2} ) ( \xi - e _ {3} ) } w . \end{equation}

For practical applications, the Jacobi form is the most suitable.

Especially important is the case when in \eqref{eq1} (or in \eqref{eq2}) $B = n ( n + 1 )$, where $n$ is a natural number. In this case the solutions of \eqref{eq1} are meromorphic in the whole plane and their properties have been thoroughly studied. Among the solutions of \eqref{eq2} with $B = n ( n + 1 )$ the Lamé functions are of paramount importance (cf. Lamé function).

References

[1] G. Lamé, "Sur les surfaces isothermes dans les corps homogènes en équilibre de température" J. Math. Pures Appl. , 2 (1837) pp. 147–188
[2] M.J.O. Strutt, "Lamésche, Mathieusche und Verwandte Funktionen in Physik und Technik" Ergebn. Math. , 1 : 3 (1932)
[3] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6
[4] H. Bateman (ed.) A. Erdélyi (ed.) , Higher transcendental functions , 3. Automorphic functions , McGraw-Hill (1955)
[5] E.W. Hobson, "The theory of spherical and ellipsoidal harmonics" , Cambridge Univ. Press (1931)
How to Cite This Entry:
Lamé equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lam%C3%A9_equation&oldid=14445
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article