# scipy.special.elliprc

scipy.special.elliprc(x, y, out=None) = <ufunc 'elliprc'>

  Degenerate symmetric elliptic integral.

  The function RC is defined as [1]

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

      Real or complex input parameters. x can be any number in the complex plane cut along the negative real axis. y must be non-zero.

    out ndarray, optional

      Optional output array for the function values

  Returns:
    R scalar or ndarray

      Value of the integral. If y is real and negative, the Cauchy principal value is returned. If both of x and y are real, the return value is real. Otherwise, the return value is complex.

  See also

  `elliprf`

    Completely-symmetric elliptic integral of the first kind.

  `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

  RC is a degenerate case of the symmetric integral RF: `elliprc(x, y) == elliprf(x, y, y)`. It is an elementary function rather than an elliptic integral.

  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.

  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.E6

  [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 elliprc

      >>> x = 1.2 + 3.4j
      >>> y = 5.
      >>> scale = 0.3 + 0.4j
      >>> elliprc(scale*x, scale*y)
      (0.5484493976710874-0.4169557678995833j)

      >>> elliprc(x, y)/np.sqrt(scale)
      (0.5484493976710874-0.41695576789958333j)

  When the two arguments coincide, the integral is particularly simple:

      >>> x = 1.2 + 3.4j
      >>> elliprc(x, x)
      (0.4299173120614631-0.3041729818745595j)

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

  Another simple case: the first argument vanishes:

      >>> y = 1.2 + 3.4j
      >>> elliprc(0, y)
      (0.6753125346116815-0.47779380263880866j)

      >>> np.pi/2/np.sqrt(y)
      (0.6753125346116815-0.4777938026388088j)

  When x and y are both positive, we can express \(R_C(x,y)\) in terms of more elementary functions. For the case \(0 \le x < y\),

      >>> x = 3.2
      >>> y = 6.
      >>> elliprc(x, y)
      0.44942991498453444

      >>> np.arctan(np.sqrt((y-x)/x))/np.sqrt(y-x)
      0.44942991498453433

  And for the case \(0 \le y < x\),

      >>> x = 6.
      >>> y = 3.2
      >>> elliprc(x,y)
      0.4989837501576147

      >>> np.log((np.sqrt(x)+np.sqrt(x-y))/np.sqrt(y))/np.sqrt(x-y)
      0.49898375015761476