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large variable and/or large parameter

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21: 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.
  • 22: 16.13 Appell Functions
    For large parameter asymptotics see López et al. (2013a, b), and Ferreira et al. (2013a, b).
    23: Bibliography T
  • N. M. Temme (1987) On the computation of the incomplete gamma functions for large values of the parameters. In Algorithms for approximation (Shrivenham, 1985), Inst. Math. Appl. Conf. Ser. New Ser., Vol. 10, pp. 479–489.
  • N. M. Temme (1994b) Computational aspects of incomplete gamma functions with large complex parameters. In Approximation and Computation. A Festschrift in Honor of Walter Gautschi, R. V. M. Zahar (Ed.), International Series of Numerical Mathematics, Vol. 119, pp. 551–562.
  • N. M. Temme (2003) Large parameter cases of the Gauss hypergeometric function. J. Comput. Appl. Math. 153 (1-2), pp. 441–462.
  • 24: 2.3 Integrals of a Real Variable
    Then … Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion: … When p ( t ) is real and x is a large positive parameter, the main contribution to the integral … When the parameter x is large the contributions from the real and imaginary parts of the integrand in … k ( ) and λ are positive constants, α is a variable parameter in an interval α 1 α α 2 with α 1 0 and 0 < α 2 k , and x is a large positive parameter. …
    25: 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.
  • 26: 8.21 Generalized Sine and Cosine Integrals
    8.21.16 Si ( a , z ) = z a k = 0 ( 2 k + 3 2 ) ( 1 - 1 2 a ) k ( 1 2 + 1 2 a ) k + 1 j 2 k + 1 ( z ) , a - 1 , - 3 , - 5 , ,
    8.21.22 f ( a , z ) = 0 sin t ( t + z ) 1 - a d t ,
    8.21.23 g ( a , z ) = 0 cos t ( t + z ) 1 - a d t .
    When | ph z | < 1 2 π , …
    §8.21(viii) Asymptotic Expansions
    27: Bibliography W
  • R. Wong (1973a) An asymptotic expansion of W k , m ( z ) with large variable and parameters. Math. Comp. 27 (122), pp. 429–436.
  • 28: 2.4 Contour Integrals
    Then … in which z is a large real or complex parameter, p ( α , t ) and q ( α , t ) are analytic functions of t and continuous in t and a second parameter α . …
    29: 25.11 Hurwitz Zeta Function
    §25.11(xii) a -Asymptotic Behavior
    Similarly, as a in the sector | ph a | 1 2 π - δ ( < 1 2 π ) , …
    30: 30.11 Radial Spheroidal Wave Functions
    30.11.3 S n m ( j ) ( z , γ ) = ( 1 - z - 2 ) 1 2 m A n - m ( γ 2 ) 2 k m - n a n , k - m ( γ 2 ) ψ n + 2 k ( j ) ( γ z ) .
    §30.11(iii) Asymptotic Behavior
    30.11.7 𝒲 { S n m ( 1 ) ( z , γ ) , S n m ( 2 ) ( z , γ ) } = 1 γ ( z 2 - 1 ) .
    30.11.11 K n m ( γ ) = π 2 ( γ 2 ) m + 1 ( - 1 ) m a n , 1 2 ( m - n + 1 ) - m ( γ 2 ) Γ ( 5 2 + m ) A n - m ( γ 2 ) ( d Ps n m ( z , γ 2 ) / d z | z = 0 ) , n - m odd.
    30.11.12 A n - m ( γ 2 ) S n m ( 1 ) ( z , γ ) = 1 2 i m + n γ m ( n - m ) ! ( n + m ) ! z m ( 1 - z - 2 ) 1 2 m - 1 1 e - i γ z t ( 1 - t 2 ) 1 2 m Ps n m ( t , γ 2 ) d t .