scipy.special.ellip_harm#

scipy.special.ellip_harm(h2, k2, n, p, s, signm=1, signn=1)[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:

h2float: h**2
k2float: k**2; should be larger than h**2
nint: Degree
sfloat: Coordinate
pint: 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:

Efloat: the harmonic \(E^p_n(s)\)

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

New in version 0.15.0.

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

Examples

>>> from scipy.special import ellip_harm
>>> w = ellip_harm(5,8,1,1,2.5)
>>> w
2.5

Check that the functions indeed are solutions to the Lame equation:

>>> import numpy as np
>>> from scipy.interpolate import UnivariateSpline
>>> def eigenvalue(f, df, ddf):
...     r = (((s**2 - h**2) * (s**2 - k**2) * ddf
...           + s * (2*s**2 - h**2 - k**2) * df
...           - n * (n + 1)*s**2*f) / f)
...     return -r.mean(), r.std()
>>> s = np.linspace(0.1, 10, 200)
>>> k, h, n, p = 8.0, 2.2, 3, 2
>>> E = ellip_harm(h**2, k**2, n, p, s)
>>> E_spl = UnivariateSpline(s, E)
>>> a, a_err = eigenvalue(E_spl(s), E_spl(s,1), E_spl(s,2))
>>> a, a_err
(583.44366156701483, 6.4580890640310646e-11)