scipy.special.ellip_harm

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

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 : float

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

See also

ellip_harm_2, ellip_normal

Notes

The geometric intepretation of the ellipsoidal functions is explained in [R302], [R303], [R304]. 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

[R301]

Digital Libary of Mathematical Functions 29.12 http://dlmf.nist.gov/29.12

[R302]

(1, 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

[R303]

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

[R304]

(1, 2) 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:

>>> 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)