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11: 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. …
12: Bibliography O
  • A. B. Olde Daalhuis and F. W. J. Olver (1995a) Hyperasymptotic solutions of second-order linear differential equations. I. Methods Appl. Anal. 2 (2), pp. 173–197.
  • A. B. Olde Daalhuis and F. W. J. Olver (1995b) On the calculation of Stokes multipliers for linear differential equations of the second order. Methods Appl. Anal. 2 (3), pp. 348–367.
  • A. B. Olde Daalhuis (1995) Hyperasymptotic solutions of second-order linear differential equations. II. Methods Appl. Anal. 2 (2), pp. 198–211.
  • A. B. Olde Daalhuis (1998a) Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one. Proc. Roy. Soc. London Ser. A 454, pp. 1–29.
  • F. W. J. Olver (1977c) Second-order differential equations with fractional transition points. Trans. Amer. Math. Soc. 226, pp. 227–241.
  • 13: 14.26 Uniform Asymptotic Expansions
    §14.26 Uniform Asymptotic Expansions
    14: 10.57 Uniform Asymptotic Expansions for Large Order
    §10.57 Uniform Asymptotic Expansions for Large Order
    15: 10.72 Mathematical Applications
    Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. … 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)). … If f ( z ) has a double zero z 0 , or more generally z 0 is a zero of order m , m = 2 , 3 , 4 , , then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order 1 / ( m + 2 ) . …The order of the approximating Bessel functions, or modified Bessel functions, is 1 / ( λ + 2 ) , except in the case when g ( z ) has a double pole at z 0 . … Then for large u asymptotic approximations of the solutions w can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on u and α ). …
    16: 14.6 Integer Order
    §14.6 Integer Order
    §14.6(i) Nonnegative Integer Orders
    §14.6(ii) Negative Integer Orders
    For connections between positive and negative integer orders see (14.9.3), (14.9.4), and (14.9.13). …
    17: 10.41 Asymptotic Expansions for Large Order
    §10.41 Asymptotic Expansions for Large Order
    §10.41(i) Asymptotic Forms
    §10.41(ii) Uniform Expansions for Real Variable
    18: 10.69 Uniform Asymptotic Expansions for Large Order
    §10.69 Uniform Asymptotic Expansions for Large Order
    19: 14.1 Special Notation
    §14.1 Special Notation
    x , y , τ real variables.
    m , n unless stated otherwise, nonnegative integers, used for order and degree, respectively.
    μ , ν general order and degree, respectively.
    20: 22.10 Maclaurin Series
    22.10.1 sn ( z , k ) = z ( 1 + k 2 ) z 3 3 ! + ( 1 + 14 k 2 + k 4 ) z 5 5 ! ( 1 + 135 k 2 + 135 k 4 + k 6 ) z 7 7 ! + O ( z 9 ) ,
    22.10.2 cn ( z , k ) = 1 z 2 2 ! + ( 1 + 4 k 2 ) z 4 4 ! ( 1 + 44 k 2 + 16 k 4 ) z 6 6 ! + O ( z 8 ) ,
    22.10.3 dn ( z , k ) = 1 k 2 z 2 2 ! + k 2 ( 4 + k 2 ) z 4 4 ! k 2 ( 16 + 44 k 2 + k 4 ) z 6 6 ! + O ( z 8 ) .
    22.10.4 sn ( z , k ) = sin z k 2 4 ( z sin z cos z ) cos z + O ( k 4 ) ,
    22.10.6 dn ( z , k ) = 1 k 2 2 sin 2 z + O ( k 4 ) ,