ellippy.misc.heuman_lambda#

ellippy.misc.heuman_lambda(phi, m)[source]#

Computes Heuman Lambda function Λ₀(φ | m).

\[\Lambda_0(\varphi, m) = \frac{F\!\left(\varphi, 1-m\right)}{K\!\left(1-m\right)} + \frac{2}{\pi} K(m)\, Z\!\left(\varphi, 1-m\right)\]
Parameters:

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

  • m (ArrayLike) – Elliptic parameter. m ∈ ℝ, 0 ≤ m < 1.

Returns:

Scalar or numpy.ndarray broadcast from inputs.

Raises:

ValueError – If m < 0 or m ≥ 1, phi is infinite, or inputs contain NaN.

Graph

Special Cases

  • Λ₀(nπ/2, m) = n where n ∈ ℤ

  • Λ0(φ, 0) = sin(φ)

Related Functions

With mc = 1 - m and Δ² = 1 - mc sin²φ:

  • Λ₀(φ, m) = F(φ, mc)/K(mc) + (2/π) K(m) Z(φ, mc)

  • Λ₀(φ, m) = 2/π · mc sin(φ) cos(φ)/Δ · [RF(0, mc, 1) + m/(3Δ²) RJ(0, mc, 1, 1 - m/Δ²)]

References

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