# 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 [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.

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 [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]

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