`scipy.special._ellip_harm:ellip_harm`

source: /scipy/special/_ellip_harm.py :7

Signature

   def     ellip_harm (    h2    ,    k2    ,    n    ,    p    ,    s    ,    signm    =  1   ,    signn    =  1     )

Summary

Ellipsoidal harmonic functions E^p_n(l)

Extended Summary

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 (2 s^{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 $E_{n}^{p} (s)$

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

Examples

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

✓

✗

Check that the functions indeed are solutions to the Lamé 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

⚠

Aliases

scipy.special.ellip_harm