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Module « scipy.special »

Fonction elliprf - module scipy.special

Signature de la fonction elliprf

def elliprf(*args, **kwargs) 

Description

help(scipy.special.elliprf)

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

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

Completely-symmetric elliptic integral of the first kind.

The function RF is defined as [1]_

.. math::

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

Parameters
----------
x, y, z : array_like
    Real or complex input parameters. `x`, `y`, or `z` can be any number in
    the complex plane cut along the negative real axis, but at most one of
    them can be 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 integral.
elliprd : Symmetric elliptic integral of the second kind.
elliprg : Completely-symmetric elliptic integral of the second kind.
elliprj : Symmetric elliptic integral of the third kind.

Notes
-----
The code implements Carlson's algorithm based on the duplication theorems
and series expansion up to the 7th order (cf.:
https://dlmf.nist.gov/19.36.i) and the AGM algorithm for the complete
integral. [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.E1
.. [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 elliprf

>>> x = 1.2 + 3.4j
>>> y = 5.
>>> z = 6.
>>> scale = 0.3 + 0.4j
>>> elliprf(scale*x, scale*y, scale*z)
(0.5328051227278146-0.4008623567957094j)

>>> elliprf(x, y, z)/np.sqrt(scale)
(0.5328051227278147-0.4008623567957095j)

All three arguments coincide:

>>> x = 1.2 + 3.4j
>>> elliprf(x, x, x)
(0.42991731206146316-0.30417298187455954j)

>>> 1/np.sqrt(x)
(0.4299173120614631-0.30417298187455954j)

The so-called "first lemniscate constant":

>>> elliprf(0, 1, 2)
1.3110287771460598

>>> from scipy.special import gamma
>>> gamma(0.25)**2/(4*np.sqrt(2*np.pi))
1.3110287771460598

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