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21: 10.40 Asymptotic Expansions for Large Argument
§10.40 Asymptotic Expansions for Large Argument
§10.40(i) Hankel’s Expansions
Products
§10.40(iv) Exponentially-Improved Expansions
22: 9.13 Generalized Airy Functions
and 𝒵 p is any linear combination of the modified Bessel functions I p and e p π i K p 10.25(ii)). Swanson and Headley (1967) define independent solutions A n ( z ) and B n ( z ) of (9.13.1) by …
- A n ( 0 ) = p 1 / 2 B n ( 0 ) = p p Γ ( p ) .
23: 32.10 Special Function Solutions
§32.10(iii) Third Painlevé Equation
24: Frank W. J. Olver
Olver joined NIST in 1961 after having been recruited by Milton Abramowitz to be the author of the Chapter “Bessel Functions of Integer Order” in the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, a publication which went on to become the most widely distributed and most highly cited publication in NIST’s history. … He is particularly known for his extensive work in the study of the asymptotic solution of differential equations, i. …, the behavior of solutions as the independent variable, or some parameter, tends to infinity, and in the study of the particular solutions of differential equations known as special functions (e. …, Bessel functions, hypergeometric functions, Legendre functions). …
  • 25: 10.68 Modulus and Phase Functions
    Equations (10.68.8)–(10.68.14) also hold with the symbols ber , bei , M , and θ replaced throughout by ker , kei , N , and ϕ , respectively. … However, numerical tabulations show that if the second of these equations applies and ϕ 1 ( x ) is continuous, then ϕ 1 ( 0 ) = - 3 4 π ; compare Abramowitz and Stegun (1964, p. 433).
    26: 30.10 Series and Integrals
    27: 30.2 Differential Equations
    If γ = 0 , Equation (30.2.4) is satisfied by spherical Bessel functions; see (10.47.1).
    28: 10.16 Relations to Other Functions
    Elementary Functions
    Parabolic Cylinder Functions
    Principal values on each side of these equations correspond.
    Confluent Hypergeometric Functions
    Generalized Hypergeometric Functions
    29: Bibliography D
  • T. M. Dunster, D. A. Lutz, and R. Schäfke (1993) Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions. Proc. Roy. Soc. London Ser. A 440, pp. 37–54.
  • T. M. Dunster (1990a) Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.
  • T. M. Dunster (2001a) Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions. Stud. Appl. Math. 107 (3), pp. 293–323.
  • 30: 2.8 Differential Equations with a Parameter
    §2.8(iv) Case III: Simple Pole
    For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv). … For further examples of uniform asymptotic approximations in terms of parabolic cylinder functions see §§13.20(iii), 13.20(iv), 14.15(v), 15.12(iii), 18.24. …