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1: Bibliography F
  • S. Farid Khwaja and A. B. Olde Daalhuis (2014) Uniform asymptotic expansions for hypergeometric functions with large parameters IV. Anal. Appl. (Singap.) 12 (6), pp. 667–710.
  • C. Ferreira, J. L. López, and E. Pérez Sinusía (2013a) The third Appell function for one large variable. J. Approx. Theory 165, pp. 60–69.
  • C. Ferreira, J. L. López, and E. Pérez Sinusía (2005) Incomplete gamma functions for large values of their variables. Adv. in Appl. Math. 34 (3), pp. 467–485.
  • J. L. Fields (1973) Uniform asymptotic expansions of certain classes of Meijer G -functions for a large parameter. SIAM J. Math. Anal. 4 (3), pp. 482–507.
  • J. L. Fields (1983) Uniform asymptotic expansions of a class of Meijer G -functions for a large parameter. SIAM J. Math. Anal. 14 (6), pp. 1204–1253.
  • 2: Bibliography B
  • G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.
  • K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.
  • L. C. Biedenharn, R. L. Gluckstern, M. H. Hull, and G. Breit (1955) Coulomb functions for large charges and small velocities. Phys. Rev. (2) 97 (2), pp. 542–554.
  • G. Blanch and I. Rhodes (1955) Table of characteristic values of Mathieu’s equation for large values of the parameter. J. Washington Acad. Sci. 45 (6), pp. 166–196.
  • A. A. Bogush and V. S. Otchik (1997) Problem of two Coulomb centres at large intercentre separation: Asymptotic expansions from analytical solutions of the Heun equation. J. Phys. A 30 (2), pp. 559–571.
  • 3: Bibliography D
  • T. M. Dunster (1986) Uniform asymptotic expansions for prolate spheroidal functions with large parameters. SIAM J. Math. Anal. 17 (6), pp. 1495–1524.
  • 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 (1991) Conical functions with one or both parameters large. Proc. Roy. Soc. Edinburgh Sect. A 119 (3-4), pp. 311–327.
  • T. M. Dunster (2003b) Uniform asymptotic expansions for associated Legendre functions of large order. Proc. Roy. Soc. Edinburgh Sect. A 133 (4), pp. 807–827.
  • A. Dzieciol, S. Yngve, and P. O. Fröman (1999) Coulomb wave functions with complex values of the variable and the parameters. J. Math. Phys. 40 (12), pp. 6145–6166.
  • 4: 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 (1975b) Legendre functions with both parameters large. Philos. Trans. Roy. Soc. London Ser. A 278, pp. 175–185.
  • F. W. J. Olver (1980b) Whittaker functions with both parameters large: Uniform approximations in terms of parabolic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 86 (3-4), pp. 213–234.
  • 5: 28.8 Asymptotic Expansions for Large q
    §28.8 Asymptotic Expansions for Large q
    Barrett’s Expansions
    The approximations apply when the parameters a and q are real and large, and are uniform with respect to various regions in the z -plane. …
    Dunster’s Approximations
    6: 12.10 Uniform Asymptotic Expansions for Large Parameter
    §12.10 Uniform Asymptotic Expansions for Large Parameter
    In this section we give asymptotic expansions of PCFs for large values of the parameter a that are uniform with respect to the variable z , when both a and z ( = x ) are real. …
    §12.10(ii) Negative a , 2 a < x <
    7: 30.9 Asymptotic Approximations and Expansions
    §30.9(i) Prolate Spheroidal Wave Functions
    For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). … For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995). … The behavior of λ n m ( γ 2 ) for complex γ 2 and large | λ n m ( γ 2 ) | is investigated in Hunter and Guerrieri (1982).
    8: 11.6 Asymptotic Expansions
    §11.6(i) Large | z | , Fixed ν
    If the series on the right-hand side of (11.6.1) is truncated after m ( 0 ) terms, then the remainder term R m ( z ) is O ( z ν 2 m 1 ) . …
    §11.6(ii) Large | ν | , Fixed z
    §11.6(iii) Large | ν | , Fixed z / ν
    For fixed λ ( > 1 )
    9: 12.11 Zeros
    If 2 n 3 2 < a < 2 n + 1 2 , n = 1 , 2 , , then U ( a , x ) has n positive real zeros. …
    §12.11(ii) Asymptotic Expansions of Large Zeros
    When a > 1 2 the zeros are asymptotically given by z a , s and z a , s ¯ , where s is a large positive integer and …
    §12.11(iii) Asymptotic Expansions for Large Parameter
    For large negative values of a the real zeros of U ( a , x ) , U ( a , x ) , V ( a , x ) , and V ( a , x ) can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …
    10: Bibliography L
  • R. E. Langer (1934) The solutions of the Mathieu equation with a complex variable and at least one parameter large. Trans. Amer. Math. Soc. 36 (3), pp. 637–695.
  • H. T. Lau (2004) A Numerical Library in Java for Scientists & Engineers. Chapman & Hall/CRC, Boca Raton, FL.
  • J. L. López, P. Pagola, and E. Pérez Sinusía (2013a) Asymptotics of the first Appell function F 1 with large parameters II. Integral Transforms Spec. Funct. 24 (12), pp. 982–999.
  • J. L. López, P. Pagola, and E. Pérez Sinusía (2013b) Asymptotics of the first Appell function F 1 with large parameters. Integral Transforms Spec. Funct. 24 (9), pp. 715–733.
  • J. L. López (1999) Asymptotic expansions of the Whittaker functions for large order parameter. Methods Appl. Anal. 6 (2), pp. 249–256.