# scipy.special.elliprg#

scipy.special.elliprg(x, y, z) = <ufunc 'elliprg'>#

Completely-symmetric elliptic integral of the second kind.

The function RG is defined as 

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

Returns
Rndarray

Value of the integral. If all of x, y, and z are real, the return value is real. Otherwise, the return value is complex.

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 

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

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. 

New 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

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

$S = 4 \pi a b c R_{\mathrm{G}}(1 / a^2, 1 / b^2, 1 / c^2).$
>>> from scipy.special import elliprg
>>> 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