# Airy functions

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##### 1: 9.1 Special Notation
 $k$ nonnegative integer, except in §9.9(iii). …
The main functions treated in this chapter are the Airy functions $\mathrm{Ai}\left(z\right)$ and $\mathrm{Bi}\left(z\right)$, and the Scorer functions $\mathrm{Gi}(z)$ and $\mathrm{Hi}(z)$ (also known as inhomogeneous Airy functions). Other notations that have been used are as follows: $\mathrm{Ai}\left(-x\right)$ and $\mathrm{Bi}\left(-x\right)$ for $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ (Jeffreys (1928), later changed to $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$); $U(x)=\sqrt{\pi}\mathrm{Bi}\left(x\right)$, $V(x)=\sqrt{\pi}\mathrm{Ai}\left(x\right)$ (Fock (1945)); $A(x)=3^{-\ifrac{1}{3}}\pi\mathrm{Ai}\left(-3^{-\ifrac{1}{3}}x\right)$ (Szegő (1967, §1.81)); $e_{0}(x)=\pi\mathrm{Hi}(-x)$, $\widetilde{e}_{0}(x)=-\pi\mathrm{Gi}(-x)$ (Tumarkin (1959)).
##### 2: 9.2 Differential Equation
###### §9.2(i) Airy’s Equation
All solutions are entire functions of $z$. …
##### 3: 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 $-\mathrm{Gi}\left(z\right),\mathrm{Ai}\left(z\right),\mathrm{Bi}\left(z\right),$ $|\operatorname{ph}z|\leq\tfrac{1}{3}\pi$,
##### 4: 9.8 Modulus and Phase
9.8.3 $M\left(x\right)=\sqrt{{\mathrm{Ai}^{2}}\left(x\right)+{\mathrm{Bi}^{2}}\left(x% \right)},$
9.8.4 $\theta\left(x\right)=\operatorname{arctan}\left(\mathrm{Ai}\left(x\right)/% \mathrm{Bi}\left(x\right)\right).$
9.8.7 $N\left(x\right)=\sqrt{{\mathrm{Ai}'^{2}}\left(x\right)+{\mathrm{Bi}'^{2}}\left% (x\right)},$
9.8.8 $\phi\left(x\right)=\operatorname{arctan}\left(\mathrm{Ai}'\left(x\right)/% \mathrm{Bi}'\left(x\right)\right).$
##### 5: 9.13 Generalized Airy Functions
###### §9.13 Generalized AiryFunctions
Swanson and Headley (1967) define independent solutions $A_{n}\left(z\right)$ and $B_{n}\left(z\right)$ of (9.13.1) by … Reid (1972) and Drazin and Reid (1981, Appendix) introduce the following contour integrals in constructing approximate solutions to the Orr–Sommerfeld equation for fluid flow: …
##### 6: 2.8 Differential Equations with a Parameter
These are elementary functions in Case I, and Airy functions9.2) in Case II. …
2.8.19 $\mathrm{Ai}\left(x\right)=\mathrm{Bi}\left(x\right)$
of smallest absolute value, and define the envelopes of $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ by
2.8.20 $\mathrm{envAi}\left(x\right)=\mathrm{envBi}\left(x\right)=\left({\mathrm{Ai}^{% 2}}\left(x\right)+{\mathrm{Bi}^{2}}\left(x\right)\right)^{1/2},$ $-\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. …
##### 7: 9.14 Incomplete Airy Functions
###### §9.14 Incomplete AiryFunctions
Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter. …
##### 8: 9.15 Mathematical Applications
###### §9.15 Mathematical Applications
Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …
##### 9: 9.16 Physical Applications
###### §9.16 Physical Applications
Airy functions are applied in many branches of both classical and quantum physics. … In fluid dynamics, Airy functions enter several topics. …An application of Airy functions to the solution of this equation is given in Gramtcheff (1981). …