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# scipy.special.ellipeinc

scipy.special.ellipeinc(phi, m, out=None) = <ufunc 'ellipeinc'>

  Incomplete elliptic integral of the second kind

  This function is defined as

  \[E(\phi, m) = \int_0^{\phi} [1 - m \sin(t)^2]^{1/2} dt\]
  Parameters:
    phi array_like

      amplitude of the elliptic integral.

    m array_like

      parameter of the elliptic integral.

    out ndarray, optional

      Optional output array for the function values

  Returns:
    E scalar or ndarray

      Value of the elliptic integral.

  See also

  `ellipkm1`

    Complete elliptic integral of the first kind, near m = 1

  `ellipk`

    Complete elliptic integral of the first kind

  `ellipkinc`

    Incomplete elliptic integral of the first kind

  `ellipe`

    Complete elliptic integral of the second kind

  `elliprd`

    Symmetric elliptic integral of the second kind.

  `elliprf`

    Completely-symmetric elliptic integral of the first kind.

  `elliprg`

    Completely-symmetric elliptic integral of the second kind.

  Notes

  Wrapper for the Cephes [1] routine ellie.

  Computation uses arithmetic-geometric means algorithm.

  The parameterization in terms of \(m\) follows that of section 17.2 in [2]. Other parameterizations in terms of the complementary parameter \(1 - m\), modular angle \(\sin^2(\alpha) = m\), or modulus \(k^2 = m\) are also used, so be careful that you choose the correct parameter.

  The Legendre E incomplete integral can be related to combinations of Carlson’s symmetric integrals R_D, R_F, and R_G in multiple ways [3]. For example, with \(c = \csc^2\phi\),

  \[E(\phi, m) = R_F(c-1, c-k^2, c) - \frac{1}{3} k^2 R_D(c-1, c-k^2, c) .\]

  References

  [1]

  Cephes Mathematical Functions Library, http://www.netlib.org/cephes/

  [2]

  Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.

  [3]

  NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.28 of 2020-09-15. See Sec. 19.25(i) https://dlmf.nist.gov/19.25#i

  Examples

  The elliptic integral of the second kind can be used to find the circumference of an ellipse with semi-major axis `a` and semi-minor axis `b`.

      >>> import numpy as np
      >>> from scipy.special import ellipeinc
      >>> a, b = 3.5, 2.1
      >>> e = np.sqrt(1.0 - b**2/a**2)  # eccentricity
      >>> 4*a*ellipeinc(np.pi/2, e**2)
      np.float64(17.86889920437869)