AI Airy Ai

AI.1 Introduction

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

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

AI.1.1

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

$\begin{equation*} \begin{split} \operatorname{Ai} (0)& =\frac{\sqrt[3]{3}}{3 \Gamma \Bigl(\frac{2}{3}\Bigr)}, \\ \frac{\partial \operatorname{Ai} (x)}{\partial x} (0)& =\frac{-\sqrt[6]{3} \Gamma \Bigl(\frac{2}{3}\Bigr)}{2 \pi}. \end{split} \end{equation*}$

AI.1.2

Related function: Airy Bi

AI.2 Series and asymptotic expansions

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AI.2.1 Taylor expansion at

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AI.2.1.1 First terms

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

AI.2.1.1.1

AI.2.1.2 General form

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

AI.2.1.2.1

The coefficients $u (n)$

satisfy the recurrence

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

AI.2.1.2.2

Initial conditions of AI.2.1.2.2 are given by

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

AI.2.1.2.3

The recurrence AI.2.1.2.2 has the closed form solution

$\begin{equation*} \begin{split} u (3 n)& =\frac{3^{\bigl(\frac{1}{3} - 2 n\bigr)}}{3 \Gamma (n + 1) \Gamma \Bigl(n + \frac{2}{3}\Bigr)}, \\ u (3 n + 1)& =\frac{-3^{\bigl(\frac{2}{3} - 2 n\bigr)}}{9 \Gamma (n + 1) \Gamma \Bigl(n + \frac{4}{3}\Bigr)}, \\ u (3 n + 2)& =0. \end{split} \end{equation*}$

AI.2.1.2.4

AI.2.2 Asymptotic expansion at $\infty$

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AI.2.2.1 First terms

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$\begin{equation*} \begin{split} & \operatorname{Ai} (x)\approx \operatorname{e} ^{\biggl(\frac{-2}{3 \xi^{3}}\biggr)} \sqrt{\xi} \Biggl(\frac{1}{2 \sqrt{\pi}} - \frac{5 \xi^{3}}{96 \sqrt{\pi}} + \ldots\Biggr) \end{split} \end{equation*}$

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

AI.2.2.2 General form

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

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

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

This recurrence has the closed form solution

$\begin{equation*} \begin{split} u (3 n)& =\frac{(-1)^{n} 6^{(2 n)} \Gamma \Bigl(n + \frac{5}{6}\Bigr) \Gamma \Bigl(n + \frac{1}{6}\Bigr)}{4 \pi^{\frac{3}{2}} 48^{n} \Gamma (n + 1)}, \\ u (3 n + 1)& =0, \\ u (3 n + 2)& =0. \end{split} \end{equation*}$

AI.3 Graphs

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AI.3.1 Real axis

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AI.3.2 Complex plane

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