scipy.special.elliprf#

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

Completely-symmetric elliptic integral of the first kind.

The function RF is defined as [1]

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

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, but at most one of them can be 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

elliprc: Degenerate symmetric integral.
elliprd: Symmetric elliptic integral of the second kind.
elliprg: Completely-symmetric elliptic integral of the second kind.
elliprj: Symmetric elliptic integral of the third kind.

Notes

The code implements Carlson’s algorithm based on the duplication theorems and series expansion up to the 7th order (cf.: https://dlmf.nist.gov/19.36.i) and the AGM algorithm for the complete integral. [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.E1

[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 elliprf

>>> x = 1.2 + 3.4j
>>> y = 5.
>>> z = 6.
>>> scale = 0.3 + 0.4j
>>> elliprf(scale*x, scale*y, scale*z)
(0.5328051227278146-0.4008623567957094j)

>>> elliprf(x, y, z)/np.sqrt(scale)
(0.5328051227278147-0.4008623567957095j)

All three arguments coincide:

>>> x = 1.2 + 3.4j
>>> elliprf(x, x, x)
(0.42991731206146316-0.30417298187455954j)

>>> 1/np.sqrt(x)
(0.4299173120614631-0.30417298187455954j)

The so-called “first lemniscate constant”:

>>> elliprf(0, 1, 2)
1.3110287771460598

>>> from scipy.special import gamma
>>> gamma(0.25)**2/(4*np.sqrt(2*np.pi))
1.3110287771460598