# scipy.special.ellipe

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

  Complete elliptic integral of the second kind

  This function is defined as

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

      Defines the 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

  `ellipeinc`

    Incomplete elliptic integral of the second kind

  `elliprd`

    Symmetric elliptic integral of the second kind.

  `elliprg`

    Completely-symmetric elliptic integral of the second kind.

  Notes

  Wrapper for the Cephes [1] routine ellpe.

  For m > 0 the computation uses the approximation,

  \[E(m) \approx P(1-m) - (1-m) \log(1-m) Q(1-m),\]

  where \(P\) and \(Q\) are tenth-order polynomials. For m < 0, the relation

  \[E(m) = E(m/(m - 1)) \sqrt(1-m)\]

  is used.

  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 integral is related to Carlson’s symmetric R_D or R_G functions in multiple ways [3]. For example,

  \[E(m) = 2 R_G(0, 1-k^2, 1) .\]

  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

  This function is used in finding the circumference of an ellipse with semi-major axis a and semi-minor axis b.

      >>> import numpy as np
      >>> from scipy import special

      >>> a = 3.5
      >>> b = 2.1
      >>> e_sq = 1.0 - b**2/a**2  # eccentricity squared

  Then the circumference is found using the following:

      >>> C = 4*a*special.ellipe(e_sq)  # circumference formula
      >>> C
      17.868899204378693

  When a and b are the same (meaning eccentricity is 0), this reduces to the circumference of a circle.

      >>> 4*a*special.ellipe(0.0)  # formula for ellipse with a = b
      21.991148575128552
      >>> 2*np.pi*a  # formula for circle of radius a
      21.991148575128552