BI Airy Bi

BI.1 Introduction

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

$\begin{equation*} \begin{split} -x y (x) + \frac{\partial^{2} y (x)}{\partial x^{2}}& =0. \end{split} \end{equation*}$

BI.1.1

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

$\begin{equation*} \begin{split} \operatorname{Bi} (0)& =\frac{3^{\frac{5}{6}}}{3 \Gamma \Bigl(\frac{2}{3}\Bigr)}, \\ \frac{\partial \operatorname{Bi} (x)}{\partial x} (0)& =\frac{3^{\frac{2}{3}} \Gamma \Bigl(\frac{2}{3}\Bigr)}{2 \pi}. \end{split} \end{equation*}$

BI.1.2

Related function: Airy Ai

BI.2 Series and asymptotic expansions

top up back next into bottom

BI.2.1 Taylor expansion at

top up back next into bottom

BI.2.1.1 First terms

top up back next into bottom

$\begin{equation*} \begin{split} \operatorname{Bi} (x)& =\frac{3^{\frac{5}{6}}}{3 \Gamma \Bigl(\frac{2}{3}\Bigr)} + \frac{3^{\frac{2}{3}} \Gamma \Bigl(\frac{2}{3}\Bigr)}{2 \pi} x + \frac{3^{\frac{5}{6}}}{18 \Gamma \Bigl(\frac{2}{3}\Bigr)} x^{3} + \frac{3^{\frac{2}{3}} \Gamma \Bigl(\frac{2}{3}\Bigr)}{24 \pi} x^{4} + \\ & \quad{}\quad{}\frac{3^{\frac{5}{6}}}{540 \Gamma \Bigl(\frac{2}{3}\Bigr)} x^{6} + \frac{3^{\frac{2}{3}} \Gamma \Bigl(\frac{2}{3}\Bigr)}{1008 \pi} x^{7} + \frac{3^{\frac{5}{6}}}{38880 \Gamma \Bigl(\frac{2}{3}\Bigr)} x^{9} + \frac{3^{\frac{2}{3}} \Gamma \Bigl(\frac{2}{3}\Bigr)}{90720 \pi} \\ & \quad{}\quad{}x^{10} + \frac{3^{\frac{5}{6}}}{5132160 \Gamma \Bigl(\frac{2}{3}\Bigr)} x^{12} + \frac{3^{\frac{2}{3}} \Gamma \Bigl(\frac{2}{3}\Bigr)}{14152320 \pi} x^{13} + \frac{3^{\frac{5}{6}}}{1077753600 \Gamma \Bigl(\frac{2}{3}\Bigr)} \\ & \quad{}\quad{}x^{15} + \operatorname{O} \bigl(x^{16}\bigr). \end{split} \end{equation*}$

BI.2.1.1.1

BI.2.1.2 General form

top up back next into bottom

$\begin{equation*} \begin{split} \operatorname{Bi} (x)& =\sum_{n = 0}^{\infty} u (n) x^{n}. \end{split} \end{equation*}$

BI.2.1.2.1

The coefficients $u (n)$

satisfy the recurrence

$\begin{equation*} \begin{split} -u (n) + \bigl(n^{2} + 5 n + 6\bigr) u (n + 3)& =0. \end{split} \end{equation*}$

BI.2.1.2.2

Initial conditions of BI.2.1.2.2 are given by

$\begin{equation*} \begin{split} u (0)& =\frac{3^{\frac{5}{6}}}{3 \Gamma \Bigl(\frac{2}{3}\Bigr)}, \\ u (1)& =\frac{3^{\frac{2}{3}} \Gamma \Bigl(\frac{2}{3}\Bigr)}{2 \pi}, \\ u (2)& =0. \end{split} \end{equation*}$

BI.2.1.2.3

The recurrence BI.2.1.2.2 has the closed form solution

$\begin{equation*} \begin{split} u (3 n + 2)& =0, \\ u (3 n)& =\frac{3^{\bigl(\frac{5}{6} - 2 n\bigr)}}{3 \Gamma (n + 1) \Gamma \Bigl(n + \frac{2}{3}\Bigr)}, \\ u (3 n + 1)& =\frac{3^{\bigl(\frac{1}{6} - 2 n\bigr)}}{3 \Gamma \Bigl(n + \frac{4}{3}\Bigr) \Gamma (n + 1)}. \end{split} \end{equation*}$

BI.2.1.2.4

BI.2.2 Asymptotic expansion at $\infty$

top up back next into bottom

BI.2.2.1 First terms

top up back next into bottom

$\begin{equation*} \begin{split} & \operatorname{Bi} (x)\approx \operatorname{e} ^{\biggl(\frac{-2}{3 \xi^{3}}\biggr)} \sqrt{\xi} \Biggl(\frac{-i}{\sqrt{\pi}} + \frac{5 i \xi^{3}}{48 \sqrt{\pi}} + \ldots\Biggr) \end{split} \end{equation*}$

where $\xi = -\sqrt{\frac{1}{x}}$

BI.2.2.2 General form

top up back next into bottom

$\begin{equation*} \begin{split} & \operatorname{Bi} (x)\approx \operatorname{e} ^{\biggl(\frac{-2}{3 \xi^{3}}\biggr)} \sqrt{\xi} \sum_{n = 0}^{\infty} u (n) \xi^{n} \end{split} \end{equation*}$

where $\xi = -\sqrt{\frac{1}{x}}$ The coefficients $u (n)$

satisfy the following recurrence

$\begin{equation*} \begin{split} 16 u (n) n + u (n - 3) \bigl(-31 + 12 n + 4 (n - 3)^{2}\bigr)& =0. \end{split} \end{equation*}$

whose initial conditions are given by

$\begin{equation*} \begin{split} u (2)& =0, \\ u (1)& =0, \\ u (0)& =\frac{-i}{\sqrt{\pi}}. \end{split} \end{equation*}$

This recurrence has the closed form solution

$\begin{equation*} \begin{split} u (3 n + 2)& =0, \\ u (3 n + 1)& =0, \\ u (3 n)& =\frac{-i(-1)^{n} 6^{(2 n)} \Gamma \Bigl(n + \frac{5}{6}\Bigr) \Gamma \Bigl(n + \frac{1}{6}\Bigr)}{2 \pi^{\frac{3}{2}} 48^{n} \Gamma (n + 1)}. \end{split} \end{equation*}$

BI.3 Graphs

top up back next into bottom

BI.3.1 Real axis

top up back next into bottom

BI.3.2 Complex plane

top up back next into bottom