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:

$$(s^2 - h^2)(s^2 - k^2)E''(s) + s(2s^2 - h^2 - k^2)E'(s) + (a - q s^2)E(s) = 0$$

where $q = (n+1)n$ and $a$ is the eigenvalue (not returned) corresponding to the solutions.

Notes

The geometric interpretation of the ellipsoidal functions is explained in , , . The signm and :None:None:`signn` arguments control the sign of prefactors for functions according to their type:

K : +1
L : signm
M : signn
N : signm*signn

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 $E^p_n(s)$

Ellipsoidal harmonic functions E^p_n(l)

See Also

ellip_harm_2

ellip_normal

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)

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

scipy.special._ellip_harm.ellip_harm scipy.special._ellip_harm.ellip_normal scipy.special._ellip_harm.ellip_harm_2

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