ellippy.legendre.ellippi#

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

Computes complete elliptic integral of the third kind Π(n | m).

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

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

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

Returns:

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

Raises:

ValueError – If n = 1, m > 1, or inputs contain NaN.

Graph

Special Cases

  • Π(0, 0) = π/2

  • Π(0, m) = K(m)

  • Π(n, 0) = π/(2sqrt{1-n}) for n < 1

  • Π(n, 0) = 0 for n > 1

  • Π(n, m) = ∞ as n → 1-

  • Π(n, 1) = sign(1-n) · ∞

  • Π(±∞, m) = 0

  • Π(n, -∞) = 0

Related Functions

  • Π(n, m) = (n/3)·RJ(0, 1 - m, 1, 1 - n) + K(m)

  • Π(n, n) = E(n)/(1 - n) for n < 1

  • Π(n, m) = K(m) - E(m)/(1 - m) for n → 1+

Notes

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

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

References

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