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in terms of Airy functions

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1: 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). …
2: 9.19 Approximations
§9.19(i) Approximations in Terms of Elementary Functions
  • Moshier (1989, §6.14) provides minimax rational approximations for calculating Ai ( x ) , Ai ( x ) , Bi ( x ) , Bi ( x ) . They are in terms of the variable ζ , where ζ = 2 3 x 3 / 2 when x is positive, ζ = 2 3 ( - x ) 3 / 2 when x is negative, and ζ = 0 when x = 0 . The approximations apply when 2 ζ < , that is, when 3 2 / 3 x < or - < x - 3 2 / 3 . The precision in the coefficients is 21S.

  • 3: 34.8 Approximations for Large Parameters
    Uniform approximations in terms of Airy functions for the 3 j and 6 j symbols are given in Schulten and Gordon (1975b). …
    4: 36.8 Convergent Series Expansions
    36.8.3 3 2 / 3 4 π 2 Ψ ( H ) ( 3 1 / 3 x ) = Ai ( x ) Ai ( y ) n = 0 ( - 3 - 1 / 3 i z ) n c n ( x ) c n ( y ) n ! + Ai ( x ) Ai ( y ) n = 2 ( - 3 - 1 / 3 i z ) n c n ( x ) d n ( y ) n ! + Ai ( x ) Ai ( y ) n = 2 ( - 3 - 1 / 3 i z ) n d n ( x ) c n ( y ) n ! + Ai ( x ) Ai ( y ) n = 1 ( - 3 - 1 / 3 i z ) n d n ( x ) d n ( y ) n ! ,
    36.8.4 Ψ ( E ) ( x ) = 2 π 2 ( 2 3 ) 2 / 3 n = 0 ( - i ( 2 / 3 ) 2 / 3 z ) n n ! ( f n ( x + i y 12 1 / 3 , x - i y 12 1 / 3 ) ) ,
    36.8.5 f n ( ζ , ζ ¯ ) = c n ( ζ ) c n ( ζ ¯ ) Ai ( ζ ) Bi ( ζ ¯ ) + c n ( ζ ) d n ( ζ ¯ ) Ai ( ζ ) Bi ( ζ ¯ ) + d n ( ζ ) c n ( ζ ¯ ) Ai ( ζ ) Bi ( ζ ¯ ) + d n ( ζ ) d n ( ζ ¯ ) Ai ( ζ ) Bi ( ζ ¯ ) ,
    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 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)). …
    §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: 18.34 Bessel Polynomials
    For uniform asymptotic expansions of y n ( x ; a ) as n in terms of Airy functions9.2) see Wong and Zhang (1997) and Dunster (2001c). …
    8: 18.15 Asymptotic Approximations
    In Terms of Airy Functions
    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). …
    9: 2.8 Differential Equations with a Parameter
    Corresponding to each positive integer n there are solutions W n , j ( u , ξ ) , j = 1 , 2 , that are C on ( α 1 , α 2 ) , and as u 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. …
    10: 33.12 Asymptotic Expansions for Large 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. …