# in terms of Airy functions

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##### 1: 9.19 Approximations
###### §9.19(i) Approximations inTerms of Elementary Functions
• Moshier (1989, §6.14) provides minimax rational approximations for calculating $\operatorname{Ai}\left(x\right)$, $\operatorname{Ai}'\left(x\right)$, $\operatorname{Bi}\left(x\right)$, $\operatorname{Bi}'\left(x\right)$. They are in terms of the variable $\zeta$, where $\zeta=\tfrac{2}{3}x^{3/2}$ when $x$ is positive, $\zeta=\tfrac{2}{3}(-x)^{3/2}$ when $x$ is negative, and $\zeta=0$ when $x=0$. The approximations apply when $2\leq\zeta<\infty$, that is, when $3^{2/3}\leq x<\infty$ or $-\infty. The precision in the coefficients is 21S.

• ##### 2: 34.8 Approximations for Large Parameters
Uniform approximations in terms of Airy functions for the $\mathit{3j}$ and $\mathit{6j}$ symbols are given in Schulten and Gordon (1975b). …
##### 3: 36.8 Convergent Series Expansions
36.8.3 $\dfrac{3^{2/3}}{4\pi^{2}}\Psi^{(\mathrm{H})}\left(3^{1/3}\mathbf{x}\right)=% \operatorname{Ai}\left(x\right)\operatorname{Ai}\left(y\right)\sum\limits_{n=0% }^{\infty}(-3^{-1/3}iz)^{n}\dfrac{c_{n}(x)c_{n}(y)}{n!}+\operatorname{Ai}\left% (x\right)\operatorname{Ai}'\left(y\right)\sum\limits_{n=2}^{\infty}(-3^{-1/3}% iz)^{n}\dfrac{c_{n}(x)d_{n}(y)}{n!}+\operatorname{Ai}'\left(x\right)% \operatorname{Ai}\left(y\right)\sum\limits_{n=2}^{\infty}(-3^{-1/3}iz)^{n}% \dfrac{d_{n}(x)c_{n}(y)}{n!}+\operatorname{Ai}'\left(x\right)\operatorname{Ai}% '\left(y\right)\sum\limits_{n=1}^{\infty}(-3^{-1/3}iz)^{n}\dfrac{d_{n}(x)d_{n}% (y)}{n!},$
36.8.4 $\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)=2\pi^{2}\left(\dfrac{2}{3}\right)^{% 2/3}\sum\limits_{n=0}^{\infty}\dfrac{\left(-i(2/3)^{2/3}z\right)^{n}}{n!}\Re% \left(f_{n}\left(\dfrac{x+iy}{12^{1/3}},\dfrac{x-iy}{12^{1/3}}\right)\right),$
36.8.5 $f_{n}(\zeta,\overline{\zeta})=c_{n}(\zeta)c_{n}(\overline{\zeta})\operatorname% {Ai}\left(\zeta\right)\operatorname{Bi}\left(\overline{\zeta}\right)+c_{n}(% \zeta)d_{n}(\overline{\zeta})\operatorname{Ai}\left(\zeta\right)\operatorname{% Bi}'\left(\overline{\zeta}\right)+d_{n}(\zeta)c_{n}(\overline{\zeta})% \operatorname{Ai}'\left(\zeta\right)\operatorname{Bi}\left(\overline{\zeta}% \right)+d_{n}(\zeta)d_{n}(\overline{\zeta})\operatorname{Ai}'\left(\zeta\right% )\operatorname{Bi}'\left(\overline{\zeta}\right),$
##### 4: 18.32 OP’s with Respect to Freud Weights
For a uniform asymptotic expansion in terms of Airy functions9.2) for the OP’s in the case $Q(x)=x^{4}$ see Bo and Wong (1999). …
##### 5: 10.72 Mathematical Applications
In regions in which (10.72.1) has a simple turning point $z_{0}$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and $z_{0}$ is a simple zero of $f(z)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\tfrac{1}{3}$9.6(i)). …
##### 6: 12.10 Uniform Asymptotic Expansions for Large Parameter
The turning points can be included if expansions in terms of Airy functions are used instead of elementary functions2.8(iii)). …
##### 7: 18.15 Asymptotic Approximations
###### InTerms of AiryFunctions
18.15.22 $L^{(\alpha)}_{n}\left(\nu x\right)=(-1)^{n}\frac{{\mathrm{e}}^{\frac{1}{2}\nu x% }}{2^{\alpha-\frac{1}{2}}x^{\frac{1}{2}\alpha+\frac{1}{4}}}\*\left(\frac{\zeta% }{x-1}\right)^{\frac{1}{4}}\left(\frac{\operatorname{Ai}\left(\nu^{\frac{2}{3}% }\zeta\right)}{\nu^{\frac{1}{3}}}\sum_{m=0}^{M-1}\frac{E_{m}(\zeta)}{\nu^{2m}}% +\frac{\operatorname{Ai}'\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{5}{3}% }}\sum_{m=0}^{M-1}\frac{F_{m}(\zeta)}{\nu^{2m}}+\operatorname{envAi}\left(\nu^% {\frac{2}{3}}\zeta\right)O\left(\frac{1}{\nu^{2M-\frac{2}{3}}}\right)\right),$
And for asymptotic expansions as $n\to\infty$ in terms of Airy functions that apply uniformly when $-1+\delta\leq t<\infty$ or $-\infty, see §§12.10(vii) and 12.10(viii). … For an error bound for the first term in the Airy-function expansions see Olver (1997b, p. 403). …
##### 8: 2.8 Differential Equations with a Parameter
Corresponding to each positive integer $n$ there are solutions $W_{n,j}(u,\xi)$, $j=1,2$, that are $C^{\infty}$ on $(\alpha_{1},\alpha_{2})$, and as $u\to\infty$For other examples of uniform asymptotic approximations and expansions of special functions in terms of Airy functions see especially §10.20 and §§12.10(vii), 12.10(viii); also §§12.14(ix), 13.20(v), 13.21(iii), 13.21(iv), 15.12(iii), 18.15(iv), 30.9(i), 30.9(ii), 32.11(ii), 32.11(iii), 33.12(i), 33.12(ii), 33.20(iv), 36.12(ii), 36.13. …
##### 9: 33.12 Asymptotic Expansions for Large $\eta$
The first set is in terms of Airy functions and the expansions are uniform for fixed $\ell$ and $\delta\leq z<\infty$, where $\delta$ is an arbitrary small positive constant. …
##### 10: 10.41 Asymptotic Expansions for Large Order
Similar analysis can be developed for the uniform asymptotic expansions in terms of Airy functions given in §10.20. …