DAI Derivative of Airy Ai

DAI.1 Introduction

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

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

Although 0 is a singularity of DAI.1.1, the initial conditions can be given by

\begin{aligned}[1] Ai ′(x) & = \frac{−\sqrt[6]{3}Γ(\frac{2}{3})}{2π}, \\ [x^{2}] Ai ′(x) & = \frac{\sqrt[3]{3}}{6Γ(\frac{2}{3})}. \end{aligned} (DAI.1.2)

Related function: Derivative of Airy Bi

DAI.2 Series and asymptotic expansions

DAI.2.1 Asymptotic expansion at ∞. DAI.2.1.1 First terms.

Ai ′(x) ≈ \frac{e^{(\frac{−2}{3ξ^{3}} )}(\frac{−1}{2\sqrt{π}} − \frac{7ξ^{3}}{96\sqrt{π}} + …)}{\sqrt{ξ}}

where ξ = \sqrt{\frac{1}{x}} DAI.2.1.2 General form.

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

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

16u(n)n + u(n − 3)(−43 + 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 + 1) & = 0, \\ u(3n + 2) & = 0, \\ u(3n) & = \frac{(−1)^{n}6^{(2n)}Γ(n + \frac{7}{6})Γ(n − \frac{1}{6})}{4π^{\frac{3}{2}} 48^{n}Γ(n + 1)}. \end{aligned}

DAI.2.2 Asymptotic expansion at 0.

DAI.2.2.1 First terms.

Ai ′(x) ≈ (\frac{\sqrt[3]{3}x^{8}}{4320Γ(\frac{2}{3})} − \frac{x^{6} \sqrt[6]{3}Γ(\frac{2}{3})}{144π} + \frac{\sqrt[3]{3}x^{5}}{90Γ(\frac{2}{3})} − \frac{x^{3} \sqrt[6]{3}Γ(\frac{2}{3})}{6π}+

\frac{\sqrt[3]{3}x^{2}}{6Γ(\frac{2}{3})} − \frac{\sqrt[6]{3}Γ(\frac{2}{3})}{2π} …).

(DAI.2.2.1.1)

DAI.2.2.2 General form. The general form of is not easy to state and requires to exhibit the basis of formal solutions of ?? (coming soon).