ellippy.misc.jacobi_zeta

ellippy.misc.jacobi_zeta(phi, m)

Computes Jacobi Zeta function Z(φ | m).

\[Z(\varphi, m) = E(\varphi, m) - \frac{E(m)\,F(\varphi, m)}{K(m)}\]

Parameters:

• phi (ArrayLike) – Amplitude angle (φ) in radians. φ ∈ ℝ.

• m (ArrayLike) – Elliptic parameter. m ∈ ℝ, m ≤ 1.

Returns:

Scalar or numpy.ndarray broadcast from inputs.

Raises:

ValueError – If any m > 1, or phi/m are infinite, or inputs contain NaN.

Graph

Special Cases

• Z(0, m) = 0

• Z(φ, 0) = 0

• Z(φ, 1) = sin(φ)·sign(cos(φ)) for φ ≠ nπ/2

• Z(nπ/2, m) = 0 for n ∈ ℤ

• Z(φ, m) = -Z(−φ, m)

Related Functions

• Z(φ, m) = E(φ, m) - E(m) F(φ, m) / K(m)

• Z(φ, m) = m sin(φ) cos(φ) √(1 - m sin²φ) · RJ(0, k_c², 1, 1 - m sin²φ) / (3 K(m))

References

• Maddock, John, Paul Bristow, Hubert Holin, and Xiaogang Zhang. “Boost Math Library: Special Functions - Elliptic Integrals.” Accessed August 30, 2025. https://www.boost.org/doc/libs/1_88_0/libs/math/doc/html/math_toolkit/ellint.html.

• Reinhardt, W. P., and P. L. Walker. “DLMF: Chapter 22 Jacobian Elliptic Functions.” Accessed August 31, 2025. https://dlmf.nist.gov/22.

• Weisstein, Eric W. “Jacobi Zeta Function.” Wolfram Research, Inc. Accessed August 31, 2025. https://mathworld.wolfram.com/JacobiZetaFunction.html.

• 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.