ellippy.bulirsch.el2#
- ellippy.bulirsch.el2(x, kc, a, b)[source]#
Computes Bulirsch incomplete integral of the second kind
el2.\[\mathrm{el2}(x, k_c, a, b) = \int_{0}^{\arctan x}\frac{a + b\,\tan^2\theta}{\sqrt{(1+\tan^2\theta)(1+k_c^2\tan^2\theta)}} \,\mathrm{d}\theta\]- Parameters:
x (
ArrayLike) – Tangent of amplitude angle, x ∈ ℝ, x = tan(φ).kc (
ArrayLike) – Complementary modulus. kc ∈ ℝ, kc ≠ 0.a (
ArrayLike) – Real-valued coefficient.b (
ArrayLike) – Real-valued coefficient.
- Returns:
Scalar or numpy.ndarray broadcast from inputs.
- Raises:
ValueError – If kc = 0 or inputs contain NaN.
Graph
Special Cases
el2(0, kc, a, b) = 0
el2(x, kc, 0, 0) = 0
el2(∞, kc, a, b) = cel2(kc, a, b)
Related Functions
- With x = tan φ and kc² = 1 - m,
F(φ, m) = el1(x, kc) = el2(x, kc, 1, 1)
E(φ, m) = el2(x, kc, 1, kc²)
el2(∞, kc, a, b) = cel2(kc, a, b)
References
Bulirsch, Roland. “Numerical Calculation of Elliptic Integrals and Elliptic Functions.” Numerische Mathematik 7, no. 1 (February 1, 1965): 78-90. https://doi.org/10.1007/BF01397975.
Carlson, B. C. “DLMF: Chapter 19 Elliptic Integrals.” Accessed February 19, 2025. https://dlmf.nist.gov/19.
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.