ellippy.legendre.ellipf

ellippy.legendre.ellipf(phi, m)

Computes incomplete elliptic integral of the first kind F(φ | m).

\[F(\varphi\,|\,m) = \int_0^{\varphi} \frac{\mathrm{d}\theta}{\sqrt{1 - m\,\sin^2\theta}}\]

Parameters:

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

• m (ArrayLike) – Elliptic parameter. m ∈ ℝ.

Returns:

Scalar or numpy.ndarray broadcast from phi and m.

Raises:

ValueError – If m sin²φ > 1 or inputs contain NaN.

Graph

Special Cases

• F(0, m) = 0

• F(φ, 0) = φ

• F(π/2, m) = K(m)

• F(φ, -∞) = 0

• F(φ, m) = sign(φ) ∞ for |φ| = ∞

Related Functions

• With c = csc²φ: F(φ, m) = RF(c - 1, c - m, c)

Notes

The elliptic modulus k is frequently used instead of the parameter m, where k² = m.

References

• Carlson, B. C. “DLMF: Chapter 19 Elliptic Integrals.” Accessed February 19, 2025. https://dlmf.nist.gov/19.

• Maddock, John, Paul Bristow, Hubert Holin, and Xiaogang Zhang. “Boost Math Library: Special Functions - Elliptic Integrals.” Accessed April 17, 2025. https://www.boost.org/doc/libs/1_88_0/libs/math/doc/html/math_toolkit/ellint.html.

• The MathWorks, Inc. “ellipticF.” Accessed April 21, 2025. https://www.mathworks.com/help/symbolic/sym.ellipticf.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.