ellippy.bulirsch.cel1#
- ellippy.bulirsch.cel1(kc)[source]#
Computes Bulirsch complete integral of the first kind
cel1.\[\mathrm{cel1}(k_c) = \int_0^{\pi/2} \frac{\mathrm{d}\theta}{\sqrt{\cos^2\theta + k_c^2\sin^2\theta}}\]- Parameters:
kc (
ArrayLike) – Complementary modulus. kc ∈ ℝ, kc ≠ 0.- Returns:
Scalar or numpy.ndarray with the same shape as kc.
- Raises:
ValueError – If kc = 0 or inputs contain NaN.
Graph
Special Cases
cel1(kc) = 0 for |kc| = ∞
Related Functions
With kc² = 1 - m: K(m) = cel(kc, 1, 1, 1) = cel1(kc)
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.