About the Project

large b

AdvancedHelp

(0.004 seconds)

11—20 of 94 matching pages

11: 28.34 Methods of Computation
  • (b)

    Use of asymptotic expansions and approximations for large q (§§28.8(i), 28.16). See also Zhang and Jin (1996, pp. 482–485).

  • (b)

    Use of asymptotic expansions and approximations for large q (§§28.8(ii)28.8(iv)).

  • 12: 15.19 Methods of Computation
    Large values of | a | or | b | , for example, delay convergence of the Gauss series, and may also lead to severe cancellation. … Initial values for moderate values of | a | and | b | can be obtained by the methods of §15.19(i), and for large values of | a | , | b | , or | c | via the asymptotic expansions of §§15.12(ii) and 15.12(iii). For example, in the half-plane z 1 2 we can use (15.12.2) or (15.12.3) to compute F ( a , b ; c + N + 1 ; z ) and F ( a , b ; c + N ; z ) , where N is a large positive integer, and then apply (15.5.18) in the backward direction. …
    13: 28.8 Asymptotic Expansions for Large q
    §28.8 Asymptotic Expansions for Large q
    §28.8(ii) Sips’ Expansions
    §28.8(iii) Goldstein’s Expansions
    Barrett’s Expansions
    14: 2.3 Integrals of a Real Variable
    Alternatively, assume b = , q ( t ) is infinitely differentiable on [ a , ) , and each of the integrals e i x t q ( s ) ( t ) d t , s = 0 , 1 , 2 , , converges as t uniformly for all sufficiently large x . … 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 …
  • (d)

    If p ( b ) = , then P 0 ( b ) = 0 and each of the integrals

    2.3.22 e i x p ( t ) P s ( t ) p ( t ) d t , s = 0 , 1 , 2 , ,

    converges at t = b uniformly for all sufficiently large x .

  • 15: Bibliography S
  • B. Simon (1982) Large orders and summability of eigenvalue perturbation theory: A mathematical overview. Int. J. Quantum Chem. 21, pp. 3–25.
  • 16: Bibliography P
  • R. B. Paris (2004) Exactification of the method of steepest descents: The Bessel functions of large order and argument. Proc. Roy. Soc. London Ser. A 460, pp. 2737–2759.
  • 17: 2.8 Differential Equations with a Parameter
    2.8.15 W n , 1 ( u , ξ ) = Ai ( u 2 / 3 ξ ) ( s = 0 n 1 A s ( ξ ) u 2 s + O ( 1 u 2 n 1 ) ) + Ai ( u 2 / 3 ξ ) ( s = 0 n 2 B s ( ξ ) u 2 s + ( 4 / 3 ) + O ( 1 u 2 n 1 ) ) ,
    2.8.16 W n , 2 ( u , ξ ) = Bi ( u 2 / 3 ξ ) ( s = 0 n 1 A s ( ξ ) u 2 s + O ( 1 u 2 n 1 ) ) + Bi ( u 2 / 3 ξ ) ( s = 0 n 2 B s ( ξ ) u 2 s + ( 4 / 3 ) + O ( 1 u 2 n 1 ) ) .
    2.8.22 W n , 1 ( u , ξ ) = Ai ( u 2 / 3 ξ ) s = 0 n 1 A s ( ξ ) u 2 s + Ai ( u 2 / 3 ξ ) s = 0 n 2 B s ( ξ ) u 2 s + ( 4 / 3 ) + envAi ( u 2 / 3 ξ ) O ( 1 u 2 n 1 ) ,
    2.8.23 W n , 2 ( u , ξ ) = Bi ( u 2 / 3 ξ ) s = 0 n 1 A s ( ξ ) u 2 s + Bi ( u 2 / 3 ξ ) s = 0 n 2 B s ( ξ ) u 2 s + ( 4 / 3 ) + envBi ( u 2 / 3 ξ ) O ( 1 u 2 n 1 ) ,
    2.8.25 W n , 1 ( u , ξ ) = ξ 1 / 2 I ν ( u ξ 1 / 2 ) s = 0 n 1 A s ( ξ ) u 2 s + ξ I ν + 1 ( u ξ 1 / 2 ) s = 0 n 2 B s ( ξ ) u 2 s + 1 + ξ 1 / 2 I ν ( u ξ 1 / 2 ) O ( 1 u 2 n 1 ) ,
    18: 13.29 Methods of Computation
    For large values of the parameters a and b the approximations in §13.8 are available. …
    19: 16.13 Appell Functions
    For large parameter asymptotics see López et al. (2013a, b), and Ferreira et al. (2013a, b).
    20: Bibliography W
  • M. I. Weinstein and J. B. Keller (1985) Hill’s equation with a large potential. SIAM J. Appl. Math. 45 (2), pp. 200–214.