ellippy.bulirsch.el3#
- ellippy.bulirsch.el3(x, kc, p)[source]#
Computes Bulirsch incomplete integral of the third kind
el3.\[\mathrm{el3}(x, k_c, p) = \int_0^{\arctan x} \frac{\mathrm{d}\theta}{\left(\cos^2\theta + p\sin^2\theta\right)\,\sqrt{\cos^2\theta + k_c^2\sin^2\theta}}\]- Parameters:
x (
ArrayLike) – Tangent of amplitude angle, x ∈ ℝ, x = tan(φ).kc (
ArrayLike) – Complementary modulus. kc ∈ ℝ, kc ≠ 0.p (
ArrayLike) – Characteristic parameter. p ∈ ℝ, p ≠ 0.
- Returns:
Scalar or numpy.ndarray broadcast from inputs. Returns the Cauchy principal value when 1 + px² < 0.
- Raises:
ValueError – If kc = 0, 1 + px² = 0, |kc| > 1 for p < 0, or inputs contain NaN.
Graph
Special Cases
el3(0, kc, p) = 0
el3(∞, kc, p) = cel(kc, p, 1, 1) = Π(1-p, 1-kc²)
Related Functions
- With x = tan φ, p = 1 - n and kc² = 1 - m:
Π(φ, n, m) = el3(x, kc, p)
el3(∞, kc, p) = cel(kc, p, 1, 1) = Π(n, m)
References
Bulirsch, R. “Numerical Calculation of Elliptic Integrals and Elliptic Functions. III.” Numerische Mathematik 13, no. 4 (August 1, 1969): 305-15. https://doi.org/10.1007/BF02165405.
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.