Elliptic integrals

The complete elliptic integrals of the first and second kind, K(m) and E(m), arise in pendulum motion, magnetic field calculations, and conformal mapping. Both are computed via the arithmetic-geometric mean (AGM) recurrence, which converges quadratically in about 10 iterations.

ellipk

Signature: (m: f32) -> f32

Computes the complete elliptic integral of the first kind, K(m) = integral(0, pi/2, dt / sqrt(1 - m*sin^2(t))). Uses K(m) = pi / (2 * AGM(1, sqrt(1-m))).

• Domain: m in [0, 1)
• At m = 1: returns +inf (logarithmic singularity)
• Outside [0, 1]: returns NaN
• Precision: ~1e-8 relative

import Nautilus.Special (ellipk)

k0 = ellipk(cast(0.0, f32))       -- pi/2 = approximately 1.5708
k_half = ellipk(cast(0.5, f32))   -- approximately 1.8541

ellipe

Signature: (m: f32) -> f32

Computes the complete elliptic integral of the second kind, E(m) = integral(0, pi/2, sqrt(1 - m*sin^2(t)) dt). Uses the AGM recurrence with an accumulated sum of squared differences.

• Domain: m in [0, 1]
• At m = 0: returns pi/2
• At m = 1: returns 1.0
• Outside [0, 1]: returns NaN
• Precision: ~1e-8 relative

import Nautilus.Special (ellipe)

e0 = ellipe(cast(0.0, f32))       -- pi/2 = approximately 1.5708
e_half = ellipe(cast(0.5, f32))   -- approximately 1.3506
e1 = ellipe(cast(1.0, f32))       -- 1.0

Edge cases

Input m	ellipk	ellipe
0.0	pi/2	pi/2
1.0	+inf	1.0
negative	NaN	NaN
> 1.0	NaN	NaN
NaN	NaN	NaN