scipy.special.elliprg#

scipy.special.elliprg(x, y, z, out=None) = <ufunc 'elliprg'>#

Completely-symmetric elliptic integral of the second kind.

The function RG is defined as [1]

\[R_{\mathrm{G}}(x, y, z) = \frac{1}{4} \int_0^{+\infty} [(t + x) (t + y) (t + z)]^{-1/2} \left(\frac{x}{t + x} + \frac{y}{t + y} + \frac{z}{t + z}\right) t dt\]

Parameters:

x, y, zarray_like: Real or complex input parameters. x, y, or z can be any number in the complex plane cut along the negative real axis.
outndarray, optional: Optional output array for the function values

Returns:

Rscalar 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.
elliprf: Completely-symmetric elliptic integral of the first kind.
elliprj: Symmetric elliptic integral of the third kind.

Notes

The implementation uses the relation [1]

\[2 R_{\mathrm{G}}(x, y, z) = z R_{\mathrm{F}}(x, y, z) - \frac{1}{3} (x - z) (y - z) R_{\mathrm{D}}(x, y, z) + \sqrt{\frac{x y}{z}}\]

and the symmetry of x, y, z when at least one non-zero parameter can be chosen as the pivot. When one of the arguments is close to zero, the AGM method is applied instead. Other special cases are computed following Ref. [2]

Added in version 1.8.0.

References

[1] (1,2)

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

[2]

B. C. Carlson, ed., Chapter 19 in “Digital Library of Mathematical Functions,” NIST, US Dept. of Commerce. https://dlmf.nist.gov/19.16.E1 https://dlmf.nist.gov/19.20.ii

Examples

Basic homogeneity property:

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

>>> x = 1.2 + 3.4j
>>> y = 5.
>>> z = 6.
>>> scale = 0.3 + 0.4j
>>> elliprg(scale*x, scale*y, scale*z)
(1.195936862005246+0.8470988320464167j)

>>> elliprg(x, y, z)*np.sqrt(scale)
(1.195936862005246+0.8470988320464165j)

Simplifications:

>>> elliprg(0, y, y)
1.756203682760182

>>> 0.25*np.pi*np.sqrt(y)
1.7562036827601817

>>> elliprg(0, 0, z)
1.224744871391589

>>> 0.5*np.sqrt(z)
1.224744871391589

The surface area of a triaxial ellipsoid with semiaxes a, b, and c is given by

\[S = 4 \pi a b c R_{\mathrm{G}}(1 / a^2, 1 / b^2, 1 / c^2).\]

>>> def ellipsoid_area(a, b, c):
...     r = 4.0 * np.pi * a * b * c
...     return r * elliprg(1.0 / (a * a), 1.0 / (b * b), 1.0 / (c * c))
>>> print(ellipsoid_area(1, 3, 5))
108.62688289491807