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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.

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====
<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>