Lamé function

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 harvs|first=Gabriel|last= Lamé|authorlink= Gabriel Lamé|year=1837. Lamé's equation appears in the method of separation of variables applied to the Laplace equation in elliptic coordinates.

Lamé's equation is : $frac{d^2y}{dx^2} = (A+Bweierp(x))y$ where "A" and "B" are constants, and $wp$ 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: $frac{d^2y}{dt^2} +frac{1}{2}left(frac{1}{t-e_1}+frac{1}{t-e_2}+frac{1}{t-e_3}
ight)frac{dy}{dt} = frac{A+Bt}{4(t-e_1)(t-e_2)(t-e_3)}y$

Lamé functions have important applications in many scientific areas, including geophysics and neuroimaging. Ellipsoidal harmonics have significant advantages over spherical harmonics for forward/inverse modeling in magnetoencephalography and electroencephalography due to superior cortical localization accuracy [ A. Irimia (2005) Electric field and potential calculation for a bioelectric current dipole in an ellipsoid. Journal of Physics A: Mathematical and General vol. 38 pp. 8123-8138 ]

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