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]

New 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:

>>> 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