Airy functions

The Airy functions Ai(x) and Bi(x) are solutions to the Airy differential equation y'' - x*y = 0. They arise in quantum mechanics (WKB approximation), optics (diffraction near caustics), and fluid dynamics.

airy_ai

Signature: [prec: {f32, f64}](x: prec) -> prec

Computes the Airy function of the first kind. For |x| <= 5, a convergent power series in x^3 is used (up to 50 terms with early termination at 1e-8 relative tolerance). For x > 5, an exponentially decaying asymptotic form exp(-2/3 * x^(3/2)) / (2*sqrt(pi) * x^(1/4)) is used.

• Domain: all reals
• Precision: f32-grade at either dtype. Above x = 5 only the leading asymptotic term is used, so f64 gains nothing there; below it, f64 is substantially better.
• Known limitation: for large negative x (x < -5), only the power series is available. The oscillatory regime works for moderate |x| but degrades for very large negative x.

import Nautilus.Special (airy_ai)

ai_0 = airy_ai(cast(0.0, f32))     -- approximately 0.3550
ai_1 = airy_ai(cast(1.0, f32))     -- approximately 0.1353
ai_neg = airy_ai(cast(-1.0, f32))  -- approximately 0.5356

airy_bi

Signature: [prec: {f32, f64}](x: prec) -> prec

Computes the Airy function of the second kind. Uses the same power series as Ai(x) for |x| <= 5, combined with a sqrt(3) scaling factor. For x > 5, an exponentially growing asymptotic form exp(2/3 * x^(3/2)) / (sqrt(pi) * x^(1/4)) is used.

• Domain: all reals
• Precision: f32-grade at either dtype above x = 5; see airy_ai.

import Nautilus.Special (airy_bi)

bi_0 = airy_bi(cast(0.0, f32))     -- approximately 0.6149
bi_1 = airy_bi(cast(1.0, f32))     -- approximately 1.2074

Edge cases

Input	airy_ai	airy_bi
0.0	0.3550	0.6149
large positive	approaches 0 (exponential decay)	grows (exponential)
large negative	oscillates (series only)	oscillates (series only)
NaN	NaN	NaN