ellippy.bulirsch.cel2

ellippy.bulirsch.cel2(kc, a, b)

Computes Bulirsch complete integral of the second kind cel2.

\[\mathrm{cel2}(k_c, a, b) = \int_{0}^{\pi/2}\frac{a + b\,\tan^2\theta}{\sqrt{(1+\tan^2\theta)(1+k_c^2\tan^2\theta)}} \,\mathrm{d}\theta\]

Parameters:

• kc (ArrayLike) – Complementary modulus. kc ∈ ℝ, kc ≠ 0.

• a (ArrayLike) – Real-valued coefficient.

• b (ArrayLike) – Real-valued coefficient.

Returns:

Scalar or numpy.ndarray with the same shape as kc.

Raises:

ValueError – If kc = 0, more than one arguments are infinite, or inputs contain NaN.

Graph

Special Cases

• cel2(kc, 0, 0) = 0

• cel(kc, a, b) = 0 for |kc| = ∞

• cel(kc, a, b) = sign(a) ∞ for |a| = ∞

• cel(kc, a, b) = sign(b) ∞ for |b| = ∞

Related Functions

• cel2(kc, a, b) = cel(kc, 1, a, b)

• With kc² = 1 - m: E(m) = cel(kc, 1, 1, kc²) = cel2(kc, 1, kc²)

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.