# Lamé equation

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

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

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:

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

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=55749
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article