Abstract
It is known that the separation of variables of the Laplace’s equation in the ellipsoidal coordinate system leads to the Lamé equation, with solutions the Lamé functions of the first and second kind, which cannot be produced analytically for a degree higher than the seventh. These compose the corresponding ellipsoidal harmonic eigenfunctions, which are necessary for the complete description and solution of physical problems in three-dimensional domains with ellipsoidal boundaries. In this work, an efficient algorithm for calculating the Lamé functions of any degree and order is proposed, adopting compact and dimensionless expressions. The developed technique has been successfully applied to important applications in electrostatics and biophysics with excellent results. In addition, important complementary issues are analyzed, regarding the derivatives of the Lamé functions, the calculation of surface integrals, the Lamé functions of the second kind and at the same time we provide a closed-form expression for the normalization constants in terms of the polynomial coefficients of the Lamé functions.
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