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# 艾里函数

## 维基百科，自由的百科全书

${\displaystyle y''-xy=0,\,\!}$

## 定义

Ai(x)（红色）和Bi(x)（蓝色）的图像

${\displaystyle \mathrm {Ai} (x)={\frac {1}{\pi ))\int _{0}^{\infty }\cos \left({\frac {t^{3)){3))+xt\right)\,dt.}$

${\displaystyle y''-xy=0.\,\!}$

${\displaystyle \mathrm {Bi} (x)={\frac {1}{\pi ))\int _{0}^{\infty }\ e^{\left(-{\frac {t^{3)){3))+xt\right)}+\sin \left({\frac {t^{3)){3))+xt\right)\,dt.}$

## 性质

${\displaystyle x=0}$时，:${\displaystyle Ai(x)}$和:${\displaystyle Bi(x)}$以及它们的导数的值为：
{\displaystyle {\begin{aligned}\mathrm {Ai} (0)&{}={\frac {1}((\sqrt[{3}]{9))\Gamma ({\frac {2}{3))))),&\quad \mathrm {Ai} '(0)&{}=-{\frac {1}((\sqrt[{3}]{3))\Gamma ({\frac {1}{3))))),\\\mathrm {Bi} (0)&{}={\frac {1}((\sqrt[{6}]{3))\Gamma ({\frac {2}{3))))),&\quad \mathrm {Bi} '(0)&{}={\frac {\sqrt[{6}]{3)){\Gamma ({\frac {1}{3))))).\end{aligned))}

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：

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

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

## 自变量是复数时的情形

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

### 图像

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

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

## 与其它特殊函数的关系

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

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

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

{\displaystyle {\begin{aligned}\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{aligned))}

## 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