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11: 2.2 Transcendental Equations
§2.2 Transcendental Equations
An important case is the reversion of asymptotic expansions for zeros of special functions. …where F 0 = f 0 and s F s ( s 1 ) is the coefficient of x - 1 in the asymptotic expansion of ( f ( x ) ) s (Lagrange’s formula for the reversion of series). …Applications to real and complex zeros of Airy functions are given in Fabijonas and Olver (1999). For other examples see de Bruijn (1961, Chapter 2).
12: 12.16 Mathematical Applications
§12.16 Mathematical Applications
In Brazel et al. (1992) exponential asymptotics are considered in connection with an eigenvalue problem involving PCFs. … Integral transforms and sampling expansions are considered in Jerri (1982).
13: 29.7 Asymptotic Expansions
§29.7 Asymptotic Expansions
§29.7(i) Eigenvalues
§29.7(ii) Lamé Functions
In Müller (1966c) it is shown how these expansions lead to asymptotic expansions for the Lamé functions Ec ν m ( z , k 2 ) and Es ν m ( z , k 2 ) . Weinstein and Keller (1985) give asymptotics for solutions of Hill’s equation (§28.29(i)) that are applicable to the Lamé equation.
14: Bibliography D
  • P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou (1999b) Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52 (11), pp. 1335–1425.
  • 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 (1996a) Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function. Proc. Roy. Soc. London Ser. A 452, pp. 1331–1349.
  • 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.
  • T. M. Dunster (2004) Convergent expansions for solutions of linear ordinary differential equations having a simple pole, with an application to associated Legendre functions. Stud. Appl. Math. 113 (3), pp. 245–270.
  • 15: 2.9 Difference Equations
    For asymptotic expansions in inverse factorial series see Olde Daalhuis (2004a). … Error bounds and applications are included. For discussions of turning points, transition points, and uniform asymptotic expansions for solutions of linear difference equations of the second order see Wang and Wong (2003, 2005). For an introduction to, and references for, the general asymptotic theory of linear difference equations of arbitrary order, see Wimp (1984, Appendix B). For applications of asymptotic methods for difference equations to orthogonal polynomials, see, e. …
    16: 6.18 Methods of Computation
    For large x and | z | , expansions in inverse factorial series (§6.10(i)) or asymptotic expansions6.12) are available. The attainable accuracy of the asymptotic expansions can be increased considerably by exponential improvement. … For an application of the Gauss–Legendre formula (§3.5(v)) see Tooper and Mark (1968). … Power series, asymptotic expansions, and quadrature can also be used to compute the functions f ( z ) and g ( z ) . … Zeros of Ci ( x ) and si ( x ) can be computed to high precision by Newton’s rule (§3.8(ii)), using values supplied by the asymptotic expansion (6.13.2) as initial approximations. …
    17: 33.12 Asymptotic Expansions for Large η
    Then, by application of the results given in §§2.8(iii) and 2.8(iv), two sets of asymptotic expansions can be constructed for F ( η , ρ ) and G ( η , ρ ) when η . …
    18: Bibliography O
  • A. B. Olde Daalhuis (2003a) Uniform asymptotic expansions for hypergeometric functions with large parameters. I. Analysis and Applications (Singapore) 1 (1), pp. 111–120.
  • A. B. Olde Daalhuis (2003b) Uniform asymptotic expansions for hypergeometric functions with large parameters. II. Analysis and Applications (Singapore) 1 (1), pp. 121–128.
  • A. B. Olde Daalhuis (2010) Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.
  • F. W. J. Olver (1965) On the asymptotic solution of second-order differential equations having an irregular singularity of rank one, with an application to Whittaker functions. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 225–243.
  • P. J. Olver (1993b) Applications of Lie Groups to Differential Equations. 2nd edition, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York.
  • 19: 10.72 Mathematical Applications
    §10.72 Mathematical Applications
    §10.72(i) Differential Equations with Turning Points
    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. … These asymptotic expansions are uniform with respect to z , including cut neighborhoods of z 0 , and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation. …
    20: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
    For asymptotic expansions of ϕ ( ρ , β ; z ) as z in various sectors of the complex z -plane for fixed real values of ρ and fixed real or complex values of β , see Wright (1935) when ρ > 0 , and Wright (1940b) when - 1 < ρ < 0 . For exponentially-improved asymptotic expansions in the same circumstances, together with smooth interpretations of the corresponding Stokes phenomenon (§§2.11(iii)2.11(v)) see Wong and Zhao (1999b) when ρ > 0 , and Wong and Zhao (1999a) when - 1 < ρ < 0 . … This reference includes exponentially-improved asymptotic expansions for E a , b ( z ) when | z | , together with a smooth interpretation of Stokes phenomena. … For incomplete modified Bessel functions and Hankel functions, including applications, see Cicchetti and Faraone (2004). …