About the Project
NIST

large a and b

AdvancedHelp

(0.006 seconds)

11—20 of 96 matching pages

11: 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.
  • 12: Bibliography Z
  • S. Zhang and J. Jin (1996) Computation of Special Functions. John Wiley & Sons Inc., New York.
  • 13: 8.18 Asymptotic Expansions of I x ( a , b )
    Large a , Fixed b
    For asymptotic expansions for large values of a and/or b of the x -solution of the equation …
    14: Bibliography N
  • M. Nardin, W. F. Perger, and A. Bhalla (1992a) Algorithm 707: CONHYP: A numerical evaluator of the confluent hypergeometric function for complex arguments of large magnitudes. ACM Trans. Math. Software 18 (3), pp. 345–349.
  • M. Nardin, W. F. Perger, and A. Bhalla (1992b) Numerical evaluation of the confluent hypergeometric function for complex arguments of large magnitudes. J. Comput. Appl. Math. 39 (2), pp. 193–200.
  • T. D. Newton (1952) Coulomb Functions for Large Values of the Parameter η . Technical report Atomic Energy of Canada Limited, Chalk River, Ontario.
  • E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
  • Numerical Recipes (commercial C, C++, Fortran 77, and Fortran 90 libraries)
  • 15: 8.12 Uniform Asymptotic Expansions for Large Parameter
    §8.12 Uniform Asymptotic Expansions for Large Parameter
    Higher coefficients A k ( χ ) , B k ( χ ) , up to k = 8 , are given in Paris (2002b). Lastly, a uniform approximation for Γ ( a , a x ) for large a , with error bounds, can be found in Dunster (1996a). …
    Inverse Function
    As a special case, …
    16: 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 …
    17: Bibliography L
  • J. L. López and P. J. Pagola (2010) The confluent hypergeometric functions M ( a , b ; z ) and U ( a , b ; z ) for large b and z . J. Comput. Appl. Math. 233 (6), pp. 1570–1576.
  • 18: Bibliography D
  • A. Deaño, J. Segura, and N. M. Temme (2010) Computational properties of three-term recurrence relations for Kummer functions. J. Comput. Appl. Math. 233 (6), pp. 1505–1510.
  • 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.
  • 19: Bibliography T
  • N. M. Temme (1986) Laguerre polynomials: Asymptotics for large degree. Technical report Technical Report AM-R8610, CWI, Amsterdam, The Netherlands.
  • 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.
  • J. Todd (1954) Evaluation of the exponential integral for large complex arguments. J. Research Nat. Bur. Standards 52, pp. 313–317.
  • P. G. Todorov (1978) Une nouvelle représentation explicite des nombres d’Euler. C. R. Acad. Sci. Paris Sér. A-B 286 (19), pp. A807–A809.
  • B. A. Troesch and H. R. Troesch (1973) Eigenfrequencies of an elliptic membrane. Math. Comp. 27 (124), pp. 755–765.
  • 20: 2.4 Contour Integrals
    Then … For large t , the asymptotic expansion of q ( t ) may be obtained from (2.4.3) by Haar’s method. This depends on the availability of a comparison function F ( z ) for Q ( z ) that has an inverse transform … Let 𝒫 denote the path for the contour integral … 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 . … For large | z | , I ( α , z ) is approximated uniformly by the integral that corresponds to (2.4.19) when f ( α , w ) is replaced by a constant. …