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.
See also
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, andcis 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