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expansions in Airy functions

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1: 9.19 Approximations
§9.19(ii) Expansions in Chebyshev Series
2: 33.20 Expansions for Small | ϵ |
§33.20(iv) Uniform Asymptotic Expansions
These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders 2 + 1 and 2 + 2 .
3: 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). …
4: 33.12 Asymptotic Expansions for Large η
Then as η , … The first set is in terms of Airy functions and the expansions are uniform for fixed and δ z < , where δ is an arbitrary small positive constant. …
5: 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. …
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)). …
§12.10(vii) Negative a , 2 a < x < . Expansions in Terms of Airy Functions
Modified Expansions
§12.10(viii) Negative a , < x < 2 a . Expansions in Terms of Airy Functions
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
18.15.22 L n ( α ) ( ν x ) = ( 1 ) n e 1 2 ν x 2 α 1 2 x 1 2 α + 1 4 ( ζ x 1 ) 1 4 ( Ai ( ν 2 3 ζ ) ν 1 3 m = 0 M 1 E m ( ζ ) ν 2 m + Ai ( ν 2 3 ζ ) ν 5 3 m = 0 M 1 F m ( ζ ) ν 2 m + envAi ( ν 2 3 ζ ) O ( 1 ν 2 M 2 3 ) ) ,
And for asymptotic expansions as n in terms of Airy functions that apply uniformly when 1 + δ t < or < t 1 δ , 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
§9.7(iii) Error Bounds for Real Variables
§9.7(iv) Error Bounds for Complex Variables
§9.7(v) Exponentially-Improved Expansions