# 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 

$R_{\mathrm{D}}(x, y, z) = \frac{3}{2} \int_0^{+\infty} [(t + x) (t + y)]^{-1/2} (t + z)^{-3/2} dt$
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.

elliprc

Degenerate symmetric elliptic integral.

elliprf

Completely-symmetric elliptic integral of the first kind.

elliprg

Completely-symmetric elliptic integral of the second kind.

elliprj

Symmetric elliptic integral of the third kind.

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. 

New in version 1.8.0.

References



B. C. Carlson, ed., Chapter 19 in “Digital Library of Mathematical Functions,” NIST, US Dept. of Commerce. https://dlmf.nist.gov/19.16.E5



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