scipy.special.ellip_harm_2

scipy.special.ellip_harm_2(h2, k2, n, p, s)

Ellipsoidal harmonic functions F^p_n(l)

These are also known as Lame functions of the second kind, and are solutions to the Lame equation:

\[(s^2 - h^2)(s^2 - k^2)F''(s) + s(2s^2 - h^2 - k^2)F'(s) + (a - q s^2)F(s) = 0\]

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.

p : int

Order, can range between [1,2n+1].

s : float

Coordinate

Returns:

F : float

The harmonic \(F^p_n(s)\)

See also

ellip_harm, ellip_normal

Notes

Lame functions of the second kind are related to the functions of the first kind:

\[F^p_n(s)=(2n + 1)E^p_n(s)\int_{0}^{1/s}\frac{du}{(E^p_n(1/u))^2\sqrt{(1-u^2k^2)(1-u^2h^2)}}\]

Examples

>>> from scipy.special import ellip_harm_2
>>> w = ellip_harm_2(5,8,2,1,10)
>>> w
0.00108056853382