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# 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, z array_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.

    out ndarray, optional

      Optional output array for the function values

  Returns:
    R scalar 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