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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:
.. math:: (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 :math:`q = (n+1)n` and :math:`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 :math:`E^p_n(s)`
See Also
--------
ellip_harm_2, ellip_normal
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
.. versionadded:: 0.15.0
References
----------
.. [1] Digital Library of Mathematical Functions 29.12
https://dlmf.nist.gov/29.12
.. [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`.
.. [3] David J.and Dechambre P, "Computation of Ellipsoidal
Gravity Field Harmonics for small solar system bodies"
pp. 30-36, 2000
.. [4] 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:
>>> 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
(583.44366156701483, 6.4580890640310646e-11)
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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:
.. math:: (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 :math:`q = (n+1)n` and :math:`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 :math:`F^p_n(s)`
Notes
-----
Lame functions of the second kind are related to the functions of the first kind:
.. math::
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)}}
.. versionadded:: 0.15.0
See Also
--------
ellip_harm, ellip_normal
Examples
--------
>>> from scipy.special import ellip_harm_2
>>> w = ellip_harm_2(5,8,2,1,10)
>>> w
0.00108056853382
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Ellipsoidal harmonic normalization constants gamma^p_n
The normalization constant is defined as
.. math::
\gamma^p_n=8\int_{0}^{h}dx\int_{h}^{k}dy\frac{(y^2-x^2)(E^p_n(y)E^p_n(x))^2}{\sqrt((k^2-y^2)(y^2-h^2)(h^2-x^2)(k^2-x^2)}
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].
Returns
-------
gamma : float
The normalization constant :math:`\gamma^p_n`
See Also
--------
ellip_harm, ellip_harm_2
Notes
-----
.. versionadded:: 0.15.0
Examples
--------
>>> from scipy.special import ellip_normal
>>> w = ellip_normal(5,8,3,7)
>>> w
1723.38796997
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