Lamé equation

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 {\mathcal p} ( z) \right ] w , \end{equation}

where $ {\mathcal p} ( z) $ is the Weierstrass ${\mathcal p}$-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 \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)