Lamé function
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In mathematics, a Lamé function (or ellipsoidal harmonic function) is a solution of Lamé's equation, a second order ordinary differential equation. It was introduced in the paper (Gabriel Lamé 1837). Lamé's equation appears in the method of separation of variables applied to the Laplace equation in elliptic coordinates.
Lamé functions are discussed in detail in (Arthur Erdélyi, Wilhelm Magnus & Fritz Oberhettinger et al. 1955, Chapter XV)
Lamé's equation is
where A and B are constants, and 
is the Weierstrass elliptic function. The most important case is when B of the form n(n + 1) for an integer n, in which case the solutions extend to meromorphic functions defined in the whole complex plane. For other values of B the solutions have branch points.
By changing the independent variable, Lamé's equation can also be rewritten in algebraic form as
[edit] References
- Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz & Tricomi, Francesco G. (1955), Higher transcendental functions. Vol. III, McGraw-Hill Book Company, Inc., New York-Toronto-London, MR0066496
- Lamé, G. (1837), “Sur les surfaces isothermes dans les corps homogènes en équilibre de température”, J. Math. Pures Appl. 2: 147–188
- Rozov, N.Kh. (2001), “Lamé equation”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- Rozov, N.Kh. (2001), “Lamé function”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
