maxframe.tensor.special.ellip_harm#

maxframe.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 Lamé functions of the first kind, and are solutions to the Lamé 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