AI Airy Ai

AI.1 Introduction

Let x be a complex variable of ℂ \ {∞}.The function Airy Ai (noted Ai) is defined by the following second order differential equation

\begin{aligned}& −xy(x) + \frac{∂^{2}y(x)}{∂x^{2}} & = 0. \\ (AI.1.1) \end{aligned}

The initial conditions of AI.1.1 are given at 0 by

\begin{aligned}Ai(0) & = \frac{\sqrt[3]{3}}{3Γ(\frac{2}{3})}, \\ \frac{∂ Ai(x)}{∂x}(0) & = \frac{−\sqrt[6]{3}Γ(\frac{2}{3})}{2π}. \end{aligned} (AI.1.2)

Related function: Airy Bi

AI.2 Series and asymptotic expansions

AI.2.1 Taylor expansion at 0. AI.2.1.1 First terms.

Ai(x) = \frac{\sqrt[3]{3}}{3Γ(\frac{2}{3})} − \frac{\sqrt[6]{3}Γ(\frac{2}{3})}{2π}x + \frac{\sqrt[3]{3}}{18Γ(\frac{2}{3})}x^{3} − \frac{\sqrt[6]{3}Γ(\frac{2}{3})}{24π}x^{4}+

\frac{\sqrt[3]{3}}{540Γ(\frac{2}{3})}x^{6} − \frac{\sqrt[6]{3}Γ(\frac{2}{3})}{1008π}x^{7} + \frac{\sqrt[3]{3}}{38880Γ(\frac{2}{3})}x^{9} − \frac{\sqrt[6]{3}Γ(\frac{2}{3})}{90720π}

x^{10} + \frac{\sqrt[3]{3}}{5132160Γ(\frac{2}{3})}x^{12} − \frac{\sqrt[6]{3}Γ(\frac{2}{3})}{14152320π} x^{13} + \frac{\sqrt[3]{3}}{1077753600Γ(\frac{2}{3})}

x^{15} + O(x^{16}).

(AI.2.1.1.1)

AI.2.1.2 General form.

\begin{aligned}& Ai(x) & = ∑_{n=0}^{∞} u(n)x^{n}. \\ (AI.2.1.2.1) \end{aligned}

The coefficients u(n) satisfy the recurrence

\begin{aligned}& −u(n) + (n^{2} + 5n + 6)u(n + 3) & = 0. \\ (AI.2.1.2.2) \end{aligned}

Initial conditions of AI.2.1.2.2 are given by

\begin{aligned}u(0) & = \frac{\sqrt[3]{3}}{3Γ(\frac{2}{3})}, \\ u(1) & = \frac{−\sqrt[6]{3}Γ(\frac{2}{3})}{2π}, \\ u(2) & = 0. \end{aligned} ; (AI.2.1.2.3)

The recurrence AI.2.1.2.2 has the closed form solution

\begin{aligned}u(3n) & = \frac{3^{(\frac{1}{3} −2n)}}{3Γ(n + 1)Γ(n + \frac{2}{3})}, \\ u(3n + 1) & = \frac{−3^{(\frac{2}{3} −2n)}}{9Γ(n + 1)Γ(n + \frac{4}{3})}, \\ u(3n + 2) & = 0. \end{aligned} ; (AI.2.1.2.4)

AI.2.2 Asymptotic expansion at ∞.

AI.2.2.1 First terms.

Ai(x) ≈ e^{(\frac{−2}{3ξ^{3}} )} \sqrt{ξ}(\frac{1}{2\sqrt{π}} − \frac{5ξ^{3}}{96\sqrt{π}} + …)

where ξ = \sqrt{\frac{1}{x}} AI.2.2.2 General form.

Ai(x) ≈ e^{(\frac{−2}{3ξ^{3}} )} \sqrt{ξ} ∑_{n=0}^{∞} u(n)ξ^{n}

where ξ = \sqrt{\frac{1}{x}}The coefficients u(n) satisfy the following recurrence

16u(n)n + u(n − 3)(−31 + 12n + 4(n − 3)^{2}) = 0.

whose initial conditions are given by

\begin{aligned}u(0) & = \frac{1}{2\sqrt{π}} , \\ u(1) & = 0, \\ u(2) & = 0. \end{aligned}

This recurrence has the closed form solution

\begin{aligned}u(3n) & = \frac{(−1)^{n}6^{(2n)}Γ(n + \frac{5}{6})Γ(n + \frac{1}{6})}{4π^{\frac{3}{2}} 48^{n}Γ(n + 1)}, \\ u(3n + 1) & = 0, \\ u(3n + 2) & = 0. \end{aligned}

AI.3 Graphs

AI.3.1 Real axis.

AI.3.2 Complex plane.