ellippy.bulirsch.cel#

ellippy.bulirsch.cel(kc, p, a, b)[source]#

Computes general complete elliptic integral in Bulirsch form cel.

\[\mathrm{cel}(k_c, p, a, b) = \int_0^{\pi/2} \frac{a\cos^2\theta + b\sin^2\theta}{(\cos^2\theta + p\sin^2\theta)\,\sqrt{\cos^2\theta + k_c^2\sin^2\theta}}\,\mathrm{d}\theta\]
Parameters:

  • kc (ArrayLike) – Complementary modulus. kc ∈ ℝ, kc ≠ 0.

  • p (ArrayLike) – Characteristic parameter. p ∈ ℝ, p ≠ 0.

  • a (ArrayLike) – Real-valued coefficient.

  • b (ArrayLike) – Real-valued coefficient.

Returns:

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

Raises:

ValueError – If kc = 0, p = 0, more than one argument is infinite, or inputs contain NaN.

Graph

Special Cases

  • cel(kc, p, 0, 0) = 0

  • cel(kc, p, a, b) = 0 for |kc| = ∞

  • cel(kc, p, a, b) = 0 for |p| = ∞

  • cel(kc, p, a, b) = sign(a) ∞ for |a| = ∞

  • cel(kc, p, a, b) = sign(b) ∞ for |b| = ∞

Related Functions

With kc² = 1 - m and p = 1 - n:

  • K(m) = cel(kc, 1, 1, 1) = cel1(kc)

  • E(m) = cel(kc, 1, 1, kc²) = cel2(kc, 1, kc²)

  • D(m) = cel(kc, 1, 0, 1)

  • Π(n, m) = cel(kc, p, 1, 1)

References

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