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Ellipsoidal Harmonics by George Dassios

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5The theory of Niven and Cartesian harmonics

5.1 The roots of the Lamé functions

In this section, we will prove that the roots of the Lamé functions are: (i) real, (ii) unequal, and (iii) belong to the interval [–h2, h2]. We will prove the results for the variable x, representing any one of the ellipsoidal variables ρ, μ, ν.

Proposition 5.1  If k(x), l(x), m(x), n(x) are Lamé functions that belong to the Lamé classes K, L, M, N, respectively, then the functions

images

do not vanish at the points x = ± h3 and x = ± h2. That is, the polynomial part of the Lamé functions have no roots at the points ±h3 and ±h2.

Proof  It is straightforward to show that ...

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