ellippy.legendre.ellipeinc#
- ellippy.legendre.ellipeinc(phi, m)[source]#
Computes incomplete elliptic integral of the second kind E(φ | m).
\[E(\varphi\,|\,m) = \int_0^{\varphi} \sqrt{1 - m\,\sin^2\theta}\,\mathrm{d}\theta\]- Parameters:
phi (
ArrayLike) – Amplitude angle (φ) in radians. φ ∈ ℝ.m (
ArrayLike) – Elliptic parameter. m ∈ ℝ.
- Returns:
Scalar or numpy.ndarray broadcast from phi and m.
- Raises:
ValueError – If m sin²φ > 1 or inputs contain NaN.
Graph
Special Cases
E(0, m) = 0
E(φ, 0) = φ
E(π/2, m) = E(m)
E(φ, 1) = sin(φ)
E(φ, -∞) = ∞
E(±∞, m) = ±∞
Related Functions
With c = csc²φ: E(φ, m) = RF(c - 1, c - m, c) - m/3 · RD(c - 1, c - m, c)
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.
The MathWorks, Inc. “ellipticE.” Accessed April 21, 2025. https://www.mathworks.com/help/symbolic/sym.elliptice.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.