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Ellipsoidal Harmonics by George Dassios

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4Ellipsoidal harmonics

4.1 Interior ellipsoidal harmonics

We consider the products of the same Lamé functions of the first kind, for the coordinates ρ, μ, v in the appropriate intervals and we define the interior ellipsoidal harmonics, or interior Lamé products, as

images

for each n = 0, 1, 2, … and m = 1, 2, …, 2n + 1, where, for points that belong to the first octant, ρ ∈ [h2, ∞), μ ∈ [h3, h2], and v ∈ [0, h3]. The functions images satisfy Laplace’s equation. We refer to the index n as the degree and to the index m as the order of the harmonic.

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