ellippy.legendre.ellipe

ellippy.legendre.ellipe(m)

Computes complete elliptic integral of the second kind E(m).

\[E(m) = \int_0^{\pi/2} \sqrt{1 - m\,\sin^2\theta}\,\mathrm{d}\theta\]

Parameters:

m (ArrayLike) – Elliptic parameter. m ∈ ℝ, m ≤ 1.

Returns:

Scalar or numpy.ndarray with the same shape as m.

Raises:

ValueError – If m > 1 or inputs contain NaN.

Graph

Special Cases

• E(0) = π/2

• E(1) = 1

• E(-∞) = ∞

Related Functions

• E(m) = 2·RG(0, 1 - m, 1).

• E(π/2, m) = E(m).

Notes

The elliptic modulus k is frequently used instead of the parameter m, where k² = m.

References

• Carlson, B. C. “DLMF: Chapter 19 Elliptic Integrals.” Accessed February 19, 2025. https://dlmf.nist.gov/19.

• Maddock, John, Paul Bristow, Hubert Holin, and Xiaogang Zhang. “Boost Math Library: Special Functions - Elliptic Integrals.” Accessed April 17, 2025. https://www.boost.org/doc/libs/1_88_0/libs/math/doc/html/math_toolkit/ellint.html.

• Abramowitz, Milton, and Irene A. Stegun. Handbook of Mathematical Functions: With Formulas, Graphs and Mathematical Tables. Unabridged, Unaltered and corr. Republ. of the 1964 ed. With Conference on mathematical tables, National science foundation, and Massachusetts institute of technology. Dover Books on Advanced Mathematics. Dover publ, 1972.

• The SciPy community. “Scipy.Special.Ellipe — SciPy v1.16.0 Manual.” Accessed July 28, 2025. https://docs.scipy.org/doc/scipy-1.16.0/reference/generated/scipy.special.ellipe.html.

• Pornsiriprasert, Sira. Ellip: Elliptic Integrals for Rust. V. 0.5.1. Released October 10, 2025. https://docs.rs/ellip/0.5.1/ellip/index.html.