# Airy function

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##### 11: Sidebar 9.SB2: Interference Patterns in Caustics
The oscillating intensity of the interference fringes across the caustic is described by the Airy function.
##### 13: 9.17 Methods of Computation
###### §9.17 Methods of Computation
Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing $\mathrm{Ai}\left(z\right)$ in the complex plane, once values of this function can be generated on the positive real axis. …
###### §9.17(v) Zeros
Zeros of the Airy functions, and their derivatives, can be computed to high precision via Newton’s rule (§3.8(ii)) or Halley’s rule (§3.8(v)), using values supplied by the asymptotic expansions of §9.9(iv) as initial approximations. … For the computation of the zeros of the Scorer functions and their derivatives see Gil et al. (2003c).
##### 14: 9.12 Scorer Functions
9.12.4 $\mathrm{Gi}\left(z\right)=\mathrm{Bi}\left(z\right)\int_{z}^{\infty}\mathrm{Ai% }\left(t\right)\mathrm{d}t+\mathrm{Ai}\left(z\right)\int_{0}^{z}\mathrm{Bi}% \left(t\right)\mathrm{d}t,$
9.12.5 $\mathrm{Hi}\left(z\right)=\mathrm{Bi}\left(z\right)\int_{-\infty}^{z}\mathrm{% Ai}\left(t\right)\mathrm{d}t-\mathrm{Ai}\left(z\right)\int_{-\infty}^{z}% \mathrm{Bi}\left(t\right)\mathrm{d}t.$
$-\mathrm{Gi}\left(x\right)$ is a numerically satisfactory companion to the complementary functions $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ on the interval $0\leq x<\infty$. …
9.12.8 $-\mathrm{Gi}\left(z\right),\mathrm{Ai}\left(z\right),\mathrm{Bi}\left(z\right),$ $|\operatorname{ph}z|\leq\tfrac{1}{3}\pi$,
##### 15: 9.4 Maclaurin Series
###### §9.4 Maclaurin Series
9.4.1 $\mathrm{Ai}\left(z\right)=\mathrm{Ai}\left(0\right)\left(1+\frac{1}{3!}z^{3}+% \frac{1\cdot 4}{6!}z^{6}+\frac{1\cdot 4\cdot 7}{9!}z^{9}+\cdots\right)+\mathrm% {Ai}'\left(0\right)\left(z+\frac{2}{4!}z^{4}+\frac{2\cdot 5}{7!}z^{7}+\frac{2% \cdot 5\cdot 8}{10!}z^{10}+\cdots\right),$
9.4.2 $\mathrm{Ai}'\left(z\right)=\mathrm{Ai}'\left(0\right)\left(1+\frac{2}{3!}z^{3}% +\frac{2\cdot 5}{6!}z^{6}+\frac{2\cdot 5\cdot 8}{9!}z^{9}+\cdots\right)+% \mathrm{Ai}\left(0\right)\left(\frac{1}{2!}z^{2}+\frac{1\cdot 4}{5!}z^{5}+% \frac{1\cdot 4\cdot 7}{8!}z^{8}+\cdots\right),$
9.4.3 $\mathrm{Bi}\left(z\right)=\mathrm{Bi}\left(0\right)\left(1+\frac{1}{3!}z^{3}+% \frac{1\cdot 4}{6!}z^{6}+\frac{1\cdot 4\cdot 7}{9!}z^{9}+\cdots\right)+\mathrm% {Bi}'\left(0\right)\left(z+\frac{2}{4!}z^{4}+\frac{2\cdot 5}{7!}z^{7}+\frac{2% \cdot 5\cdot 8}{10!}z^{10}+\cdots\right),$
9.4.4 $\mathrm{Bi}'\left(z\right)=\mathrm{Bi}'\left(0\right)\left(1+\frac{2}{3!}z^{3}% +\frac{2\cdot 5}{6!}z^{6}+\frac{2\cdot 5\cdot 8}{9!}z^{9}+\cdots\right)+% \mathrm{Bi}\left(0\right)\left(\frac{1}{2!}z^{2}+\frac{1\cdot 4}{5!}z^{5}+% \frac{1\cdot 4\cdot 7}{8!}z^{8}+\cdots\right).$
##### 17: 9.6 Relations to Other Functions
###### §9.6(iii) AiryFunctions as Confluent Hypergeometric Functions
To express Airy functions in terms of hypergeometric functions combine §9.6(i) with (10.39.9).
##### 19: 9.5 Integral Representations
###### §9.5(i) Real Variable
9.5.2 $\mathrm{Ai}\left(-x\right)=\frac{x^{\ifrac{1}{2}}}{\pi}\int_{-1}^{\infty}\cos% \left(x^{\ifrac{3}{2}}(\tfrac{1}{3}t^{3}+t^{2}-\tfrac{2}{3})\right)\mathrm{d}t,$ $x>0$.
###### §9.5(ii) Complex Variable
9.5.6 $\mathrm{Ai}\left(z\right)=\frac{\sqrt{3}}{2\pi}\int_{0}^{\infty}\exp\left(-% \frac{t^{3}}{3}-\frac{z^{3}}{3t^{3}}\right)\mathrm{d}t,$ $|\operatorname{ph}z|<\tfrac{1}{6}\pi$.
##### 20: 32.5 Integral Equations
32.5.1 $K(z,\zeta)=k\mathrm{Ai}\left(\frac{z+\zeta}{2}\right)+\frac{k^{2}}{4}\*\int_{z% }^{\infty}\!\!\!\int_{z}^{\infty}K(z,s)\mathrm{Ai}\left(\frac{s+t}{2}\right)% \mathrm{Ai}\left(\frac{t+\zeta}{2}\right)\mathrm{d}s\mathrm{d}t,$
32.5.3 $w(z)\sim k\mathrm{Ai}\left(z\right),$ $z\to+\infty$.