ellippy.bulirsch.el1#
- ellippy.bulirsch.el1(x, kc)[source]#
Computes Bulirsch incomplete integral of the first kind
el1.\[\mathrm{el1}(x, k_c) = \int_0^{\arctan x} \frac{\mathrm{d}\theta}{\sqrt{\cos^2\theta + k_c^2\sin^2\theta}}\]- Parameters:
x (
ArrayLike) – Tangent of amplitude angle, x ∈ ℝ, x = tan(φ).kc (
ArrayLike) – Complementary modulus. kc ∈ ℝ, kc ≠ 0.
- Returns:
Scalar or numpy.ndarray broadcast from inputs.
- Raises:
ValueError – If kc = 0 or inputs contain NaN.
Graph
Special Cases
el1(0, kc) = 0
el1(∞, kc) = cel1(kc)
el1(x, ∞) = 0
Related Functions
- With x = tan φ and kc² = 1 - m:
F(φ, m) = el1(x, kc) = el2(x, kc, 1, 1)
el1(∞, kc) = 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.