# 艾里函数

$y'' - xy = 0 , \,\!$

## 定义

Ai(x)（红色）和Bi(x)（蓝色）的图像

$\mathrm{Ai}(x) = \frac{1}{\pi} \int_0^\infty \cos\left(\frac{t^3}{3} + xt\right)\, dt.$

$y'' - xy = 0 . \,\!$

$\mathrm{Bi}(x) = \frac{1}{\pi} \int_0^\infty \ e^{\left(-\frac{t^3}{3} + xt\right)} + \sin\left(\frac{t^3}{3} + xt\right)\,dt.$

## 性质

$x = 0$时，:$Ai(x)$和:$Bi(x)$以及它们的导数的值为：
\begin{align} \mathrm{Ai}(0) &{}= \frac{1}{\sqrt[3]{9} \Gamma(\frac23)}, & \quad \mathrm{Ai}'(0) &{}= -\frac{1}{\sqrt[3]{3} \Gamma(\frac13)}, \\ \mathrm{Bi}(0) &{}= \frac{1}{\sqrt[6]{3}\Gamma(\frac23)}, & \quad \mathrm{Bi}'(0) &{}= \frac{\sqrt[6]{3}}{\Gamma(\frac13)}. \end{align}

x是正数时，Ai(x)是正的凸函数，指数衰减为零，Bi(x)也是正的凸函数，但呈指数增长。当x是负数时，Ai(x)和Bi(x)在零附近振动，其频率逐渐上升，振幅逐渐下降。这可以由以下艾里函数的渐近公式推出。

## 渐近公式

x趋于+∞时，艾里函数的渐近表现为：

\begin{align} \mathrm{Ai}(x) &{}\sim \frac{e^{-\frac23x^{3/2}}}{2\sqrt\pi\,x^{1/4}} \\ \mathrm{Bi}(x) &{}\sim \frac{e^{\frac23x^{3/2}}}{\sqrt\pi\,x^{1/4}}. \end{align}

\begin{align} \mathrm{Ai}(-x) &{}\sim \frac{\sin(\frac23x^{3/2}+\frac14\pi)}{\sqrt\pi\,x^{1/4}} \\ \mathrm{Bi}(-x) &{}\sim \frac{\cos(\frac23x^{3/2}+\frac14\pi)}{\sqrt\pi\,x^{1/4}}. \end{align}

## 自变量是复数时的情形

$\mathrm{Ai}(z) = \frac{1}{2\pi i} \int_{C} \exp\left(\frac{t^3}{3} - zt\right)\, dt,$

### 图像

$\Re \left[ \mathrm{Ai} ( x + iy) \right]$ $\Im \left[ \mathrm{Ai} ( x + iy) \right]$ $| \mathrm{Ai} ( x + iy) | \,$ $\mathrm{arg} \left[ \mathrm{Ai} ( x + iy) \right] \,$

$\Re \left[ \mathrm{Bi} ( x + iy) \right]$ $\Im \left[ \mathrm{Bi} ( x + iy) \right]$ $| \mathrm{Bi} ( x + iy) | \,$ $\mathrm{arg} \left[ \mathrm{Bi} ( x + iy) \right] \,$

## 与其它特殊函数的关系

\begin{align} \mathrm{Ai}(x) &{}= \frac1\pi \sqrt{\frac13 x} \, K_{1/3}\left(\frac23 x^{3/2}\right), \\ \mathrm{Bi}(x) &{}= \sqrt{\frac13 x} \left(I_{1/3}\left(\frac23 x^{3/2}\right) + I_{-1/3}\left(\frac23 x^{3/2}\right)\right). \end{align}

\begin{align} \mathrm{Ai}(-x) &{}= \frac13 \sqrt{x} \left(J_{1/3}\left(\frac23 x^{3/2}\right) + J_{-1/3}\left(\frac23 x^{3/2}\right)\right), \\ \mathrm{Bi}(-x) &{}= \sqrt{\frac13 x} \left(J_{-1/3}\left(\frac23 x^{3/2}\right) - J_{1/3}\left(\frac23 x^{3/2}\right)\right). \end{align}

Scorer函数$y'' - xy = 1/\pi$的解，它也可以用艾里函数来表示：

\begin{align} \mathrm{Gi}(x) &{}= \mathrm{Bi}(x) \int_x^\infty \mathrm{Ai}(t) \, dt + \mathrm{Ai}(x) \int_0^x \mathrm{Bi}(t) \, dt, \\ \mathrm{Hi}(x) &{}= \mathrm{Bi}(x) \int_{-\infty}^x \mathrm{Ai}(t) \, dt - \mathrm{Ai}(x) \int_{-\infty}^x \mathrm{Bi}(t) \, dt. \end{align}

## 参考文献

1. ^ 参看Abramowitz and Stegun, 1954 和 Olver, 1974。
• Milton Abramowitz and Irene A. Stegun (1954). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, (See §10.4). National Bureau of Standards.
• Airy (1838). On the intensity of light in the neighbourhood of a caustic. Transactions of the Cambridge Philosophical Society, 6, 379–402.
• Olver (1974). Asymptotics and Special Functions, Chapter 11. Academic Press, New York.
• Harold Richard Suiter. Star Testing Astronomical Telescopes: A Manual for Optical Evaluation and Adjustment. Richmond, VA: Willmann-Bell. 1994. ISBN 978-0-943396-44-6.含有许多图像