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r""" 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 http://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:
>>> 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)
""" return _ellip_harm(h2, k2, n, p, s, signm, signn)
r""" 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
""" with np.errstate(all='ignore'): return _ellip_harm_2_vec(h2, k2, n, p, s)
return _ellipsoid_norm(h2, k2, n, p)
r""" 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
""" with np.errstate(all='ignore'): return _ellip_normal_vec(h2, k2, n, p) |