BI Airy Bi

BI.1 Introduction

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

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

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

\begin{aligned}Bi(0) & = \frac{3^{\frac{5}{6}}}{3Γ(\frac{2}{3})}, \\ \frac{∂ Bi(x)}{∂x}(0) & = \frac{3^{\frac{2}{3}} Γ(\frac{2}{3})}{2π}. \end{aligned} (BI.1.2)

Related function: Airy Ai

BI.2 Series and asymptotic expansions

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

Bi(x) = \frac{3^{\frac{5}{6}}}{3Γ(\frac{2}{3})} + \frac{3^{\frac{2}{3}} Γ(\frac{2}{3})}{2π}x + \frac{3^{\frac{5}{6}}}{18Γ(\frac{2}{3})}x^{3} + \frac{3^{\frac{2}{3}} Γ(\frac{2}{3})}{24π}x^{4}+

\frac{3^{\frac{5}{6}}}{540Γ(\frac{2}{3})}x^{6} + \frac{3^{\frac{2}{3}} Γ(\frac{2}{3})}{1008π}x^{7} + \frac{3^{\frac{5}{6}}}{38880Γ(\frac{2}{3})}x^{9} + \frac{3^{\frac{2}{3}} Γ(\frac{2}{3})}{90720π}

x^{10} + \frac{3^{\frac{5}{6}}}{5132160Γ(\frac{2}{3})}x^{12} + \frac{3^{\frac{2}{3}} Γ(\frac{2}{3})}{14152320π} x^{13} + \frac{3^{\frac{5}{6}}}{1077753600Γ(\frac{2}{3})}

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

(BI.2.1.1.1)

BI.2.1.2 General form.

\begin{aligned}& Bi(x) & = ∑_{n=0}^{∞} u(n)x^{n}. \\ (BI.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. \\ (BI.2.1.2.2) \end{aligned}

Initial conditions of BI.2.1.2.2 are given by

\begin{aligned}u(0) & = \frac{3^{\frac{5}{6}}}{3Γ(\frac{2}{3})}, \\ u(1) & = \frac{3^{\frac{2}{3}} Γ(\frac{2}{3})}{2π}, \\ u(2) & = 0. \end{aligned} ; (BI.2.1.2.3)

The recurrence BI.2.1.2.2 has the closed form solution

\begin{aligned}u(3n + 2) & = 0, \\ u(3n) & = \frac{3^{(\frac{5}{6} −2n)}}{3Γ(n + 1)Γ(n + \frac{2}{3})}, \\ u(3n + 1) & = \frac{3^{(\frac{1}{6} −2n)}}{3Γ(n + \frac{4}{3})Γ(n + 1)}. \end{aligned} ; (BI.2.1.2.4)

BI.2.2 Asymptotic expansion at ∞.

BI.2.2.1 First terms.

Bi(x) ≈ e^{(\frac{−2}{3ξ^{3}} )} \sqrt{ξ}(\frac{−i}{\sqrt{π}} + \frac{5iξ^{3}}{48\sqrt{π}} + …)

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

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

This recurrence has the closed form solution

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

BI.3 Graphs

BI.3.1 Real axis.

BI.3.2 Complex plane.