A linear ordinary second-order differential equation in the complex domain
where is the Weierstrass -function and and are constants. This equation was first studied by G. Lamé ; 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:
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:
For practical applications the Jacobi form is the most suitable.
Especially important is the case when in (1) (or (2)) , where 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 the Lamé functions are of paramount importance (cf. Lamé function).
|||G. Lamé, "Sur les surfaces isothermes dans les corps homogènes en équilibre de température" J. Math. Pures Appl. , 2 (1837) pp. 147–188|
|||M.J.O. Strutt, "Lamésche, Mathieusche und Verwandte Funktionen in Physik und Technik" Ergebn. Math. , 1 : 3 (1932)|
|||E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6|
|||H. Bateman (ed.) A. Erdélyi (ed.) , Higher transcendental functions , 3. Automorphic functions , McGraw-Hill (1955)|
|||E.W. Hobson, "The theory of spherical and ellipsoidal harmonics" , Cambridge Univ. Press (1931)|
Lamé equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lam%C3%A9_equation&oldid=23366