Difference between revisions of "Lamé equation"
From Encyclopedia of Mathematics
(Importing text file) |
(details) |
||
| (4 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
| + | <!-- | ||
| + | l0574001.png | ||
| + | $#A+1 = 11 n = 0 | ||
| + | $#C+1 = 11 : ~/encyclopedia/old_files/data/L057/L.0507400 Lam\Aee equation | ||
| + | Automatically converted into TeX, above some diagnostics. | ||
| + | Please remove this comment and the {{TEX|auto}} line below, | ||
| + | if TeX found to be correct. | ||
| + | --> | ||
| + | |||
| + | {{TEX|auto}}{{TEX|done}} | ||
| + | |||
A linear ordinary second-order differential equation in the complex domain | 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 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*} | ||
| + | \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. | + | For practical applications, the Jacobi form is the most suitable. |
| − | Especially important is the case when in ( | + | 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, | + | <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
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