ellippy.legendre.ellippiinc#

ellippy.legendre.ellippiinc(n, phi, m)[source]#

Computes incomplete elliptic integral of the third kind Π(n; φ | m).

\[\Pi(n;\,\varphi\,|\,m) = \int_0^{\varphi} \frac{\mathrm{d}\theta}{\left(1 - n\,\sin^2\theta\right)\,\sqrt{1 - m\,\sin^2\theta}}\]
Parameters:

  • n (ArrayLike) – Characteristic. n ∈ ℝ, n ≠ 1.

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

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

Returns:

Scalar or numpy.ndarray broadcast from n, phi, and m. Returns the Cauchy principal value if n sin²φ > 1.

Raises:

ValueError – If m sin²φ > 1, n sin²φ = 1, m ≥ 1 with φ not a multiple of π/2, or inputs contain NaN.

Graph

Special Cases

  • Π(0, n, m) = 0

  • Π(φ, 0, 0) = φ

  • Π(φ, 1, 0) = tan(φ)

  • Π(φ, 0, m) = F(φ, m)

Related Functions

  • Π(φ, 0, m) = F(φ, m)

  • With c = csc²φ: Π(φ, n, m) = (n/3) · RJ(c - 1, c - m, c, c - n) + F(φ, m)

  • With x = tan φ, p = 1 - n, and k_c² = 1 - m: Π(φ, n, m) = el3(x, k_c, p)

Notes

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

  • The characteristic n is sometimes expressed in term of α, where α² = n.

  • This function ellippiinc (of the package) is the circular or hyperbolic case of Π, since n and m are real. It is called circular if n·(n - m)·(n - 1) is negative and hyperbolic if it is positive.

References

On this page

This Page