expansions in Airy functions

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1: 9.19 Approximations

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§9.19(ii) Expansions in Chebyshev Series

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2: 33.20 Expansions for Small $|ϵ|$

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§33.20(iv) Uniform Asymptotic Expansions

… ►These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders $2\mathrm{\ell }+1$ and $2\mathrm{\ell }+2$.

3: 33.12 Asymptotic Expansions for Large $\eta$

… ►Then as $\eta \to \mathrm{\infty }$, … ►The first set is in terms of Airy functions and the expansions are uniform for fixed $\mathrm{\ell }$ and , where $\delta$ is an arbitrary small positive constant. …

4: 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. …

5: 18.32 OP’s with Respect to Freud Weights

… ►For a uniform asymptotic expansion in terms of Airy functions (§9.2) for the OP’s in the case $Q(x)=x^4$ see Bo and Wong (1999). …

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 functions (§2.8(iii)). … ►

§12.10(vii) Negative $a$, . Expansions in Terms of Airy Functions

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Modified Expansions

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§12.10(viii) Negative $a$, . Expansions in Terms of Airy Functions

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7: 30.9 Asymptotic Approximations and Expansions

… ►For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). … ►For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995). …

8: 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. …

9: 18.15 Asymptotic Approximations

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18.15.22 $$L_n^{(\alpha )}\left(\nu x\right)={(-1)}^n\frac{{\mathrm{e}}^{\frac{1}{2}\nu x}}{2^{\alpha -\frac{1}{2}}{x}^{\frac{1}{2}\alpha +\frac{1}{4}}}{\left(\frac{\zeta }{x-1}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Ai}\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\sum _{m=0}^{M-1}\frac{E_m(\zeta )}{{\nu }^{2m}}+\frac{{\mathrm{Ai}}^{\prime }\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\sum _{m=0}^{M-1}\frac{F_m(\zeta )}{{\nu }^{2m}}+\mathrm{envAi}\left({\nu }^{\frac{2}{3}}\zeta \right)O\left(\frac{1}{{\nu }^{2M-\frac{2}{3}}}\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\sim$: Poincaré asymptotic expansion, $\mathrm{envAi}\left(x\right)$: envelope of Airy function $\mathrm{Ai}\left(x\right)$ , $\mathrm{e}$: base of natural logarithm, $m$: nonnegative integer, $n$: nonnegative integer, $\nu$ , $\zeta$ , $E_m$: coefficient, $F_m$: coefficient and $x$: real variable

Referenced by:

Erratum (V1.0.11) for Equation (18.15.22)

Permalink:

http://dlmf.nist.gov/18.15.E22

Encodings:

pMML, png, TeX

Errata (effective with 1.0.11):

Originally this equation was expressed in terms of the asymptotic symbol $\sim$. As a consequence of the use of the $O$ order symbol on the right-hand side, $\sim$ was replaced by $=$.

See also:

Annotations for §18.15(iv), §18.15(iv), §18.15 and Ch.18

… ►And for asymptotic expansions as $n\to \mathrm{\infty }$ in terms of Airy functions that apply uniformly when or , 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). …

10: 9.7 Asymptotic Expansions

§9.7 Asymptotic Expansions

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§9.7(iii) Error Bounds for Real Variables

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§9.7(iv) Error Bounds for Complex Variables

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§9.7(v) Exponentially-Improved Expansions

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