mars.tensor.special.ellip_harm_2#

mars.tensor.special.ellip_harm_2(h2, k2, n, p, s, **kwargs)[source]#

Ellipsoidal harmonic functions F^p_n(l)

These are also known as Lame functions of the second kind, and are solutions to the Lame equation:

\[(s^2 - h^2)(s^2 - k^2)F''(s) + s(2s^2 - h^2 - k^2)F'(s) + (a - q s^2)F(s) = 0\]

where \(q = (n+1)n\) and \(a\) is the eigenvalue (not returned) corresponding to the solutions.

Parameters

  • h2 (float) – h**2

  • k2 (float) – k**2; should be larger than h**2

  • n (int) – Degree.

  • p (int) – Order, can range between [1,2n+1].

  • s (float) – Coordinate

Returns

F – The harmonic \(F^p_n(s)\)

Return type

float

Notes

Lame functions of the second kind are related to the functions of the first kind:

\[F^p_n(s)=(2n + 1)E^p_n(s)\int_{0}^{1/s}\frac{du}{(E^p_n(1/u))^2\sqrt{(1-u^2k^2)(1-u^2h^2)}}\]

See also

ellip_harm, ellip_normal