DBI Derivative of Airy Bi

DBI.1 Introduction

Let x be a complex variable of ℂ \ {∞}.The function Derivative of Airy Bi (noted Bi ′) 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. \\ (DBI.1.1) \end{aligned}

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

\begin{aligned}[1] Bi ′(x) & = \frac{3^{\frac{2}{3}} Γ(\frac{2}{3})}{2π}, \\ [x^{2}] Bi ′(x) & = \frac{3^{\frac{5}{6}}}{6Γ(\frac{2}{3})}. \end{aligned} (DBI.1.2)

Related function: Derivative of Airy Ai

DBI.2 Series and asymptotic expansions

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

Bi ′(x) ≈ \frac{e^{(\frac{−2}{3ξ^{3}} )}(\frac{i}{\sqrt{π}} + \frac{7iξ^{3}}{48\sqrt{π}} + …)}{\sqrt{ξ}}

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

Bi ′(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{i}{\sqrt{π}} , \\ u(2) & = 0, \\ u(1) & = 0. \end{aligned}

This recurrence has the closed form solution

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

DBI.2.2 Asymptotic expansion at 0.

DBI.2.2.1 First terms.

Bi ′(x) ≈ (\frac{3^{\frac{5}{6}} x^{8}}{4320Γ(\frac{2}{3})} + \frac{x^{6}3^{\frac{2}{3}} Γ(\frac{2}{3})}{144π} + \frac{3^{\frac{5}{6}} x^{5}}{90Γ(\frac{2}{3})} + \frac{x^{3}3^{\frac{2}{3}} Γ(\frac{2}{3})}{6π}+

\frac{3^{\frac{5}{6}} x^{2}}{6Γ(\frac{2}{3})} + \frac{3^{\frac{2}{3}} Γ(\frac{2}{3})}{2π} …).

(DBI.2.2.1.1)

DBI.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).