scipy.special.elliprd(x, y, z, out=None) = <ufunc 'elliprd'>#

Symmetric elliptic integral of the second kind.

The function RD is defined as [1]

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

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


Degenerate symmetric elliptic integral.


Completely-symmetric elliptic integral of the first kind.


Completely-symmetric elliptic integral of the second kind.


Symmetric elliptic integral of the third kind.


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]

New in version 1.8.0.



B. C. Carlson, ed., Chapter 19 in “Digital Library of Mathematical Functions,” NIST, US Dept. of Commerce.


B. C. Carlson, “Numerical computation of real or complex elliptic integrals,” Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995.


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)
>>> elliprd(x, y, z)*np.power(scale, -1.5)

All three arguments coincide:

>>> x = 1.2 + 3.4j
>>> elliprd(x, x, x)
>>> np.power(x, -1.5)

The so-called “second lemniscate constant”:

>>> elliprd(0, 2, 1)/3
>>> from scipy.special import gamma
>>> gamma(0.75)**2/np.sqrt(2*np.pi)