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11: 10.39 Relations to Other Functions
§10.39 Relations to Other Functions
Airy Functions
Parabolic Cylinder Functions
Confluent Hypergeometric Functions
Generalized Hypergeometric Functions and Hypergeometric Function
12: 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. … , 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. …Having witnessed the birth of the computer age firsthand (as a colleague of Alan Turing at NPL, for example), Olver is also well known for his contributions to the development and analysis of numerical methods for computing special functions. … In April 2011, NIST co-organized a conference on “Special Functions in the 21st Century: Theory & Application” which was dedicated to Olver. …
  • 13: Bibliography V
  • G. Valent (1986) An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.
  • O. Vallée and M. Soares (2010) Airy Functions and Applications to Physics. Second edition, Imperial College Press, London.
  • C. G. van der Laan and N. M. Temme (1984) Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-Like Functions. CWI Tract, Vol. 10, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam.
  • R. Vidūnas and N. M. Temme (2002) Symbolic evaluation of coefficients in Airy-type asymptotic expansions. J. Math. Anal. Appl. 269 (1), pp. 317–331.
  • H. Volkmer (1982) Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.
  • 14: 9 Airy and Related Functions
    Chapter 9 Airy and Related Functions
    15: Bibliography N
  • National Bureau of Standards (1958) Integrals of Airy Functions. National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..
  • National Bureau of Standards (1967) Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..
  • J. Negro, L. M. Nieto, and O. Rosas-Ortiz (2000) Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.
  • L. N. Nosova and S. A. Tumarkin (1965) Tables of Generalized Airy Functions for the Asymptotic Solution of the Differential Equations ϵ ( p y ) + ( q + ϵ r ) y = f . Pergamon Press, Oxford.
  • Number Theory Web (website)
  • 16: 18.34 Bessel Polynomials
    §18.34(i) Definitions and Recurrence Relation
    Because the coefficients C n in (18.34.4) are not all positive, the polynomials y n ( x ; a ) cannot be orthogonal on the line with respect to a positive weight function. … where primes denote derivatives with respect to x . … 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). …
    17: 16.18 Special Cases
    §16.18 Special Cases
    This is a consequence of the following relations: …As a corollary, special cases of the F 1 1 and F 1 2 functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer G -function. Representations of special functions in terms of the Meijer G -function are given in Erdélyi et al. (1953a, §5.6), Luke (1969a, §§6.4–6.5), and Mathai (1993, §3.10).
    18: Bibliography G
  • W. Gautschi (1959b) Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.
  • W. Gautschi (1984) Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.
  • W. Gautschi (2002a) Computation of Bessel and Airy functions and of related Gaussian quadrature formulae. BIT 42 (1), pp. 110–118.
  • A. G. Gibbs (1973) Problem 72-21, Laplace transforms of Airy functions. SIAM Rev. 15 (4), pp. 796–798.
  • T. V. Gramtcheff (1981) An application of Airy functions to the Tricomi problem. Math. Nachr. 102 (1), pp. 169–181.
  • 19: Bibliography M
  • A. J. MacLeod (1994) Computation of inhomogeneous Airy functions. J. Comput. Appl. Math. 53 (1), pp. 109–116.
  • P. Martín, R. Pérez, and A. L. Guerrero (1992) Two-point quasi-fractional approximations to the Airy function Ai ( x ) . J. Comput. Phys. 99 (2), pp. 337–340.
  • F. Matta and A. Reichel (1971) Uniform computation of the error function and other related functions. Math. Comp. 25 (114), pp. 339–344.
  • G. J. Miel (1981) Evaluation of complex logarithms and related functions. SIAM J. Numer. Anal. 18 (4), pp. 744–750.
  • S. C. Milne (1985c) A new symmetry related to SU ( n ) for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.
  • 20: 18.15 Asymptotic Approximations
    In Terms of Airy Functions
    Here Ai denotes the Airy function9.2), Ai denotes its derivative, and envAi denotes its envelope (§2.8(iii)). … 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). With μ = 2 n + 1 the expansions in Chapter 12 are for the parabolic cylinder function U ( - 1 2 μ 2 , μ t 2 ) , which is related to the Hermite polynomials via … For an error bound for the first term in the Airy-function expansions see Olver (1997b, p. 403). …