About the Project
NIST

Dunster’s

AdvancedHelp

(0.001 seconds)

1—10 of 17 matching pages

1: 28.8 Asymptotic Expansions for Large q
Dunsters Approximations
Dunster (1994a) supplies uniform asymptotic approximations for numerically satisfactory pairs of solutions of Mathieu’s equation (28.2.1). …
2: Bibliography D
  • T. M. Dunster, R. B. Paris, and S. Cang (1998) On the high-order coefficients in the uniform asymptotic expansion for the incomplete gamma function. Methods Appl. Anal. 5 (3), pp. 223–247.
  • T. M. Dunster (1989) Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.
  • 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 (2014) Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point. Anal. Appl. (Singap.) 12 (4), pp. 385–402.
  • 3: 8.22 Mathematical Applications
    For further information on ζ x ( s ) , including zeros and uniform asymptotic approximations, see Kölbig (1970, 1972a) and Dunster (2006). …
    4: 10.49 Explicit Formulas
    See also §18.34, de Bruin et al. (1981a, b), and Dunster (2001c). …
    §10.49(iii) Rayleigh’s Formulas
    10.49.17 s k ( n + 1 2 ) = ( 2 k ) ! ( n + k ) ! 2 2 k ( k ! ) 2 ( n - k ) ! , k = 0 , 1 , , n .
    10.49.18 j n 2 ( z ) + y n 2 ( z ) = k = 0 n s k ( n + 1 2 ) z 2 k + 2 .
    10.49.20 ( i n ( 1 ) ( z ) ) 2 - ( i n ( 2 ) ( z ) ) 2 = ( - 1 ) n + 1 k = 0 n ( - 1 ) k s k ( n + 1 2 ) z 2 k + 2 .
    5: Staff
  • Howard S. Cohl, Technical Editor, NIST

  • T. Mark Dunster, San Diego State University, Chap. 14

  • Roderick S. C. Wong, City University of Hong Kong, Chaps. 1, 2, 18

  • T. Mark Dunster, San Diego State University, for Chap. 14

  • Roderick S. C. Wong, City University of Hong Kong, for Chaps. 2, 18

  • 6: Preface
  • T. Mark Dunster, San Diego State University

  • Boggs, S. … S. … S. … S. …
    7: 2.8 Differential Equations with a Parameter
    For another approach to these problems based on convergent inverse factorial series expansions see Dunster et al. (1993) and Dunster (2001a, 2004). … For error bounds, more delicate error estimates, extensions to complex ξ , ν , and u , zeros, and examples see Olver (1997b, Chapter 12), Boyd (1990a), and Dunster (1990a). … For two coalescing turning points see Olver (1975a, 1976) and Dunster (1996a); in this case the uniform approximants are parabolic cylinder functions. … For a coalescing turning point and double pole see Boyd and Dunster (1986) and Dunster (1990b); in this case the uniform approximants are Bessel functions of variable order. For a coalescing turning point and simple pole see Nestor (1984) and Dunster (1994b); in this case the uniform approximants are Whittaker functions (§13.14(i)) with a fixed value of the second parameter. …
    8: 2.11 Remainder Terms; Stokes Phenomenon
    with … Application of Watson’s lemma (§2.4(i)) yields … See also Paris and Kaminski (2001, Chapter 6), and Dunster (1996b, 1997). … For error bounds see Dunster (1996c). …
    9: 18.15 Asymptotic Approximations
    18.15.1 ( sin 1 2 θ ) α + 1 2 ( cos 1 2 θ ) β + 1 2 P n ( α , β ) ( cos θ ) = π - 1 2 2 n + α + β + 1 B ( n + α + 1 , n + β + 1 ) ( m = 0 M - 1 f m ( θ ) 2 m ( 2 n + α + β + 2 ) m + O ( n - M ) ) ,
    18.15.3 C m , ( α , β ) = ( 1 2 + α ) ( 1 2 - α ) ( 1 2 + β ) m - ( 1 2 - β ) m - ,
    For large β , fixed α , and 0 n / β c , Dunster (1999) gives asymptotic expansions of P n ( α , β ) ( z ) that are uniform in unbounded complex z -domains containing z = ± 1 . … The asymptotic behavior of the classical OP’s as x ± with the degree and parameters fixed is evident from their explicit polynomial forms; see, for example, (18.2.7) and the last two columns of Table 18.3.1. … See also Dunster (1999), Atia et al. (2014) and Temme (2015, Chapter 32).
    10: 14.15 Uniform Asymptotic Approximations
    14.15.3 Q ν μ ( x ) = 1 μ ν + ( 1 / 2 ) ( π u 2 ) 1 / 2 I ν + 1 2 ( μ u ) ( 1 + O ( 1 μ ) ) ,
    For asymptotic expansions and explicit error bounds, see Dunster (2003b) and Gil et al. (2000). … For asymptotic expansions and explicit error bounds, see Dunster (2003b). … For convergent series expansions see Dunster (2004). … For asymptotic expansions and explicit error bounds, see Boyd and Dunster (1986). …