ellip_harm — SciPy v1.15.2 Manual (original) (raw)

scipy.special.

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 \(E^p_n(s)\)

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

Added in version 0.15.0.

References

[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 = (((s2 - h2) * (s2 - k2) * ddf ... + s * (2*s2 - h2 - k2) * df ... - n * (n + 1)s**2f) / 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(h2, 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)