ellip_harm(h2, k2, n, p, s, signm=1, signn=1)
These are also known as Lame functions of the first kind, and are solutions to the Lame equation:
where q = (n+1)n and a is the eigenvalue (not returned) corresponding to the solutions.
The geometric interpretation of the ellipsoidal functions is explained in , , . 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
h**2
k**2; should be larger than h**2
Degree
Coordinate
Order, can range between [1,2n+1]
Sign of prefactor of functions. Can be +/-1. See Notes.
Sign of prefactor of functions. Can be +/-1. See Notes.
the harmonic E^p_n(s)
Ellipsoidal harmonic functions E^p_n(l)
from scipy.special import ellip_harm
w = ellip_harm(5,8,1,1,2.5)
w
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
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