# Airy functions

## Definitions

Symbol: AiryAi $\operatorname{Ai}\!\left(z\right)$ Airy function of the first kind
Symbol: AiryBi $\operatorname{Bi}\!\left(z\right)$ Airy function of the second kind

## Illustrations

Image: X-ray of $\operatorname{Ai}\!\left(z\right)$ on $z \in \left[-6, 6\right] + \left[-6, 6\right] i$
Image: X-ray of $\operatorname{Bi}\!\left(z\right)$ on $z \in \left[-6, 6\right] + \left[-6, 6\right] i$

## Differential equation

$y''(z) - z y(z) = 0\; \text{ where } y(z) = C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right)$
$\operatorname{Ai}\!\left(z\right) \operatorname{Bi}'\!\left(z\right) - \operatorname{Ai}'\!\left(z\right) \operatorname{Bi}\!\left(z\right) = \frac{1}{\pi}$

## Special values

$\operatorname{Ai}\!\left(0\right) = \frac{1}{{3}^{2 / 3} \Gamma\!\left(\frac{2}{3}\right)} \in \left[0.355028053887817239260063186004 \pm 1.84 \cdot 10^{-31}\right]$
$\operatorname{Ai}'\!\left(0\right) = -\frac{1}{{3}^{1 / 3} \Gamma\!\left(\frac{1}{3}\right)} \in \left[-0.258819403792806798405183560189 \pm 2.04 \cdot 10^{-31}\right]$
$\operatorname{Bi}\!\left(0\right) = \frac{1}{{3}^{1 / 6} \Gamma\!\left(\frac{2}{3}\right)} \in \left[0.614926627446000735150922369094 \pm 3.87 \cdot 10^{-31}\right]$
$\operatorname{Bi}'\!\left(0\right) = \frac{{3}^{1 / 6}}{\Gamma\!\left(\frac{1}{3}\right)} \in \left[0.448288357353826357914823710399 \pm 1.72 \cdot 10^{-31}\right]$

## Higher derivatives

$\operatorname{Ai}''\!\left(z\right) = z \operatorname{Ai}\!\left(z\right)$
$\operatorname{Bi}''\!\left(z\right) = z \operatorname{Bi}\!\left(z\right)$
${y}^{(n)}(z) = z {y}^{(n - 2)}(z) + \left(n - 2\right) {y}^{(n - 3)}(z)\; \text{ where } y(z) = C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right)$

## Hypergeometric representations

$\operatorname{Ai}\!\left(z\right) = \operatorname{Ai}\!\left(0\right) \,{}_0F_1\!\left(\frac{2}{3}, \frac{{z}^{3}}{9}\right) + z \operatorname{Ai}'\!\left(0\right) \,{}_0F_1\!\left(\frac{4}{3}, \frac{{z}^{3}}{9}\right)$
$\operatorname{Bi}\!\left(z\right) = \operatorname{Bi}\!\left(0\right) \,{}_0F_1\!\left(\frac{2}{3}, \frac{{z}^{3}}{9}\right) + z \operatorname{Bi}'\!\left(0\right) \,{}_0F_1\!\left(\frac{4}{3}, \frac{{z}^{3}}{9}\right)$
$\operatorname{Ai}'\!\left(z\right) = \operatorname{Ai}'\!\left(0\right) \,{}_0F_1\!\left(\frac{1}{3}, \frac{{z}^{3}}{9}\right) + \frac{{z}^{2}}{2} \operatorname{Ai}\!\left(0\right) \,{}_0F_1\!\left(\frac{5}{3}, \frac{{z}^{3}}{9}\right)$
$\operatorname{Bi}'\!\left(z\right) = \operatorname{Bi}'\!\left(0\right) \,{}_0F_1\!\left(\frac{1}{3}, \frac{{z}^{3}}{9}\right) + \frac{{z}^{2}}{2} \operatorname{Bi}\!\left(0\right) \,{}_0F_1\!\left(\frac{5}{3}, \frac{{z}^{3}}{9}\right)$

## Analytic properties

$C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right) \text{ is holomorphic on } z \in \mathbb{C}$
$\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} \left[C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right)\right] = \left\{\right\}$
$\operatorname{EssentialSingularities}\!\left(C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}\right\}$
$\operatorname{BranchPoints}\!\left(C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}$
$\operatorname{BranchCuts}\!\left(C \operatorname{Ai}\!\left(z\right) + D \operatorname{Bi}\!\left(z\right), z, \mathbb{C}\right) = \left\{\right\}$
$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \operatorname{Ai}\!\left(z\right) \subset \mathbb{R}$
$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \operatorname{Ai}'\!\left(z\right) \subset \mathbb{R}$

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-04-08 16:14:44.404316 UTC