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Module « scipy.special »

Fonction elliprd - module scipy.special

Signature de la fonction elliprd

def elliprd(*args, **kwargs) 

Description

help(scipy.special.elliprd)

elliprd(x1, x2, x3, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])

elliprd(x, y, z, out=None)

Symmetric elliptic integral of the second kind.

The function RD is defined as [1]_

.. math::

    R_{\mathrm{D}}(x, y, z) =
       \frac{3}{2} \int_0^{+\infty} [(t + x) (t + y)]^{-1/2} (t + z)^{-3/2}
       dt

Parameters
----------
x, y, z : array_like
    Real or complex input parameters. `x` or `y` can be any number in the
    complex plane cut along the negative real axis, but at most one of them
    can be zero, while `z` must be non-zero.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
R : scalar or ndarray
    Value of the integral. If all of `x`, `y`, and `z` are real, the
    return value is real. Otherwise, the return value is complex.

See Also
--------
elliprc : Degenerate symmetric elliptic integral.
elliprf : Completely-symmetric elliptic integral of the first kind.
elliprg : Completely-symmetric elliptic integral of the second kind.
elliprj : Symmetric elliptic integral of the third kind.

Notes
-----
RD is a degenerate case of the elliptic integral RJ: ``elliprd(x, y, z) ==
elliprj(x, y, z, z)``.

The code implements Carlson's algorithm based on the duplication theorems
and series expansion up to the 7th order. [2]_

.. versionadded:: 1.8.0

References
----------
.. [1] B. C. Carlson, ed., Chapter 19 in "Digital Library of Mathematical
       Functions," NIST, US Dept. of Commerce.
       https://dlmf.nist.gov/19.16.E5
.. [2] B. C. Carlson, "Numerical computation of real or complex elliptic
       integrals," Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995.
       https://arxiv.org/abs/math/9409227
       https://doi.org/10.1007/BF02198293

Examples
--------
Basic homogeneity property:

>>> import numpy as np
>>> from scipy.special import elliprd

>>> x = 1.2 + 3.4j
>>> y = 5.
>>> z = 6.
>>> scale = 0.3 + 0.4j
>>> elliprd(scale*x, scale*y, scale*z)
(-0.03703043835680379-0.24500934665683802j)

>>> elliprd(x, y, z)*np.power(scale, -1.5)
(-0.0370304383568038-0.24500934665683805j)

All three arguments coincide:

>>> x = 1.2 + 3.4j
>>> elliprd(x, x, x)
(-0.03986825876151896-0.14051741840449586j)

>>> np.power(x, -1.5)
(-0.03986825876151894-0.14051741840449583j)

The so-called "second lemniscate constant":

>>> elliprd(0, 2, 1)/3
0.5990701173677961

>>> from scipy.special import gamma
>>> gamma(0.75)**2/np.sqrt(2*np.pi)
0.5990701173677959

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