ellippy.legendre.ellipd

ellippy.legendre.ellipd(m)

Computes complete elliptic integral of Legendre’s type D(m).

\[D(m) = \int_0^{\pi/2} \frac{\sin^2\theta}{\sqrt{1 - m\,\sin^2\theta}}\,\mathrm{d}\theta\]

Parameters:

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

Returns:

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

Raises:

ValueError – If m > 1 or inputs contain NaN.

Graph

Special Cases

• D(0) = π/4

• D(1) = ∞

• D(-∞) = 0

Related Functions

• D(m) = (K(m) - E(m)) / m

• D(m) = RD(0, 1 - m, 1) / 3

• D(π/2, m) = D(m)

Notes

The elliptic modulus k is frequently used instead of 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.

• 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.