Airy functions

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1: 9.1 Special Notation

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$k$	nonnegative integer, except in §9.9(iii).
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►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)); $A(x)=3^{-1/3}\pi \mathrm{Ai}\left(-3^{-1/3}x\right)$ (Szegő (1967, §1.81)); $e_0(x)=\pi \mathrm{Hi}(-x)$, ${\tilde{e}}_0(x)=-\pi \mathrm{Gi}(-x)$ (Tumarkin (1959)).

2: 9.14 Incomplete Airy Functions

§9.14 Incomplete Airy Functions

►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. …

3: 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. …

4: 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). ►Airy functions play a prominent role in problems defined by nonlinear wave equations. …

5: 9.3 Graphics

§9.3 Graphics

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§9.3(i) Real Variable

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Figure 9.3.6: ${\mathrm{Bi}}^{\prime }\left(x+\mathrm{i}y\right)$. Magnify 3D Help

6: 9.19 Approximations

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§9.19(i) Approximations in Terms of Elementary Functions

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Martín et al. (1992) provides two simple formulas for approximating $\mathrm{Ai}\left(x\right)$ to graphical accuracy, one for , the other for .

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§9.19(ii) Expansions in Chebyshev Series

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Razaz and Schonfelder (1980) covers $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$. The Chebyshev coefficients are given to 30D.

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§9.19(iii) Approximations in the Complex Plane

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7: 9.8 Modulus and Phase

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§9.8(i) Definitions

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§9.8(ii) Identities

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§9.8(iii) Monotonicity

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§9.8(iv) Asymptotic Expansions

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8: 9.2 Differential Equation

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§9.2(i) Airy’s Equation

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§9.2(ii) Initial Values

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§9.2(iv) Wronskians

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§9.2(v) Connection Formulas

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9: Sidebar 9.SB1: Supernumerary Rainbows

… ►For each color, the intensity profile across the rainbow is an Airy function. Airy invented his function in 1838 precisely to describe this phenomenon more accurately than Young had done in 1800 when pointing out that supernumerary rainbows require the wave theory of light and are impossible to explain with Newton’s picture of light as a stream of independent corpuscles. …

10: 9.11 Products

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§9.11(i) Differential Equation

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§9.11(ii) Wronskian

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§9.11(iii) Integral Representations

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§9.11(iv) Indefinite Integrals

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§9.11(v) Definite Integrals

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