About the Project

in inverse factorial series

AdvancedHelp

(0.003 seconds)

1—10 of 19 matching pages

1: 6.18 Methods of Computation
For large x and | z | , expansions in inverse factorial series6.10(i)) or asymptotic expansions (§6.12) are available. …
2: 10.41 Asymptotic Expansions for Large Order
For expansions in inverse factorial series see Dunster et al. (1993). …
3: 2.9 Difference Equations
For asymptotic expansions in inverse factorial series see Olde Daalhuis (2004a). …
4: 6.10 Other Series Expansions
§6.10(i) Inverse Factorial Series
5: 2.8 Differential Equations with a Parameter
In Case I there are no transition points in 𝐃 and g ( z ) is analytic. … The same approach is used in all three cases. … These are elementary functions in Case I, and Airy functions (§9.2) in Case II. In Case III the approximating equation is … For another approach to these problems based on convergent inverse factorial series expansions see Dunster et al. (1993) and Dunster (2001a, 2004). …
6: 4.24 Inverse Trigonometric Functions: Further Properties
§4.24 Inverse Trigonometric Functions: Further Properties
§4.24(i) Power Series
§4.24(ii) Derivatives
§4.24(iii) Addition Formulas
The above equations are interpreted in the sense that every value of the left-hand side is a value of the right-hand side and vice versa. …
7: Bibliography O
  • O. M. Ogreid and P. Osland (1998) Summing one- and two-dimensional series related to the Euler series. J. Comput. Appl. Math. 98 (2), pp. 245–271.
  • A. B. Olde Daalhuis (2004a) Inverse factorial-series solutions of difference equations. Proc. Edinb. Math. Soc. (2) 47 (2), pp. 421–448.
  • J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.
  • F. W. J. Olver (1994a) Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations. Methods Appl. Anal. 1 (1), pp. 1–13.
  • F. W. J. Olver (1994b) The Generalized Exponential Integral. In Approximation and Computation (West Lafayette, IN, 1993), R. V. M. Zahar (Ed.), International Series of Numerical Mathematics, Vol. 119, pp. 497–510.
  • 8: 18.18 Sums
    Legendre
    Laguerre
    Hermite
    Ultraspherical
    Hermite
    9: 24.19 Methods of Computation
    For number-theoretic applications it is important to compute B 2 n ( mod p ) for 2 n p 3 ; in particular to find the irregular pairs ( 2 n , p ) for which B 2 n 0 ( mod p ) . We list here three methods, arranged in increasing order of efficiency.
  • Tanner and Wagstaff (1987) derives a congruence ( mod p ) for Bernoulli numbers in terms of sums of powers. See also §24.10(iii).

  • Buhler et al. (1992) uses the expansion

    24.19.3 t 2 cosh t 1 = 2 n = 0 ( 2 n 1 ) B 2 n t 2 n ( 2 n ) ! ,

    and computes inverses modulo p of the left-hand side. Multisectioning techniques are applied in implementations. See also Crandall (1996, pp. 116–120).

  • A method related to “Stickelberger codes” is applied in Buhler et al. (2001); in particular, it allows for an efficient search for the irregular pairs ( 2 n , p ) . Discrete Fourier transforms are used in the computations. See also Crandall (1996, pp. 120–124).

  • 10: 4.38 Inverse Hyperbolic Functions: Further Properties
    §4.38 Inverse Hyperbolic Functions: Further Properties
    §4.38(i) Power Series
    §4.38(ii) Derivatives
    In the following equations square roots have their principal values. …
    §4.38(iii) Addition Formulas