scipy.special.elliprd#
- scipy.special.elliprd(x, y, z, out=None) = <ufunc 'elliprd'>#
- Symmetric elliptic integral of the second kind. - The function RD is defined as [1] \[R_{\mathrm{D}}(x, y, z) = \frac{3}{2} \int_0^{+\infty} [(t + x) (t + y)]^{-1/2} (t + z)^{-3/2} dt\]- Parameters:
- x, y, zarray_like
- Real or complex input parameters. x or y can be any number in the complex plane cut along the negative real axis, but at most one of them can be zero, while z must be non-zero. 
- 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 - Notes - RD is a degenerate case of the elliptic integral RJ: - elliprd(x, y, z) == elliprj(x, y, z, z).- The code implements Carlson’s algorithm based on the duplication theorems and series expansion up to the 7th order. [2] - Added in version 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.E5 [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 elliprd - >>> x = 1.2 + 3.4j >>> y = 5. >>> z = 6. >>> scale = 0.3 + 0.4j >>> elliprd(scale*x, scale*y, scale*z) (-0.03703043835680379-0.24500934665683802j) - >>> elliprd(x, y, z)*np.power(scale, -1.5) (-0.0370304383568038-0.24500934665683805j) - All three arguments coincide: - >>> x = 1.2 + 3.4j >>> elliprd(x, x, x) (-0.03986825876151896-0.14051741840449586j) - >>> np.power(x, -1.5) (-0.03986825876151894-0.14051741840449583j) - The so-called “second lemniscate constant”: - >>> elliprd(0, 2, 1)/3 0.5990701173677961 - >>> from scipy.special import gamma >>> gamma(0.75)**2/np.sqrt(2*np.pi) 0.5990701173677959