mars.tensor.special.ellip_harm#

mars.tensor.special.ellip_harm(h2, k2, n, p, s, signm=1, signn=1, **kwargs)[source]#

Ellipsoidal harmonic functions E^p_n(l)

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

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

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

  • s (float) – Coordinate

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

  • signm ({1, -1}, optional) – Sign of prefactor of functions. Can be +/-1. See Notes.

  • signn ({1, -1}, optional) – Sign of prefactor of functions. Can be +/-1. See Notes.

Returns

E – the harmonic \(E^p_n(s)\)

Return type

float

See also

ellip_harm_2, ellip_normal

Notes

The geometric interpretation of the ellipsoidal functions is explained in 2, 3, 4. The signm and signn arguments control the sign of prefactors for functions according to their type:

K : +1
L : signm
M : signn
N : signm*signn

References

1

Digital Library of Mathematical Functions 29.12 https://dlmf.nist.gov/29.12

2

Bardhan and Knepley, “Computational science and re-discovery: open-source implementations of ellipsoidal harmonics for problems in potential theory”, Comput. Sci. Disc. 5, 014006 (2012) :doi:`10.1088/1749-4699/5/1/014006`.

3

David J.and Dechambre P, “Computation of Ellipsoidal Gravity Field Harmonics for small solar system bodies” pp. 30-36, 2000

4

George Dassios, “Ellipsoidal Harmonics: Theory and Applications” pp. 418, 2012