ellippy.bulirsch.cel3#
- ellippy.bulirsch.cel3(kc, p)[source]#
Computes complete elliptic integral of the third kind in Bulirsch’s form
cel3.\[\mathrm{cel3}(k_c, p) = \int_0^{\pi/2} \frac{\mathrm{d}\theta}{\left(\cos^2\theta + p\sin^2\theta\right)\,\sqrt{\cos^2\theta + k_c^2\sin^2\theta}}\]- Parameters:
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 p < 0.
- Raises:
ValueError – If kc = 0, p = 0, more than one argument is infinite, or inputs contain NaN.
Special Cases
cel3(kc, 1) = cel1(kc)
cel3(kc, p) = 0 for |kc| = ∞
cel3(kc, p) = 0 for |p| = ∞
Related Functions
- With kc² = 1 - m and p = 1 - n:
Π(n, m) = cel(kc, p, 1, 1) = cel3(kc, p)
cel3(kc, 1) = cel1(kc) = K(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.
Pornsiriprasert, Sira. Ellip: Elliptic Integrals for Rust. V. 1.2.0. Released October 2, 2026. https://docs.rs/ellip/1.2.0/ellip/index.html.