About the Project
NIST

Stieltjes constants

AdvancedHelp

(0.002 seconds)

9 matching pages

1: 25.2 Definition and Expansions
25.2.4 ζ ( s ) = 1 s - 1 + n = 0 ( - 1 ) n n ! γ n ( s - 1 ) n ,
where the Stieltjes constants γ n are defined via
25.2.5 γ n = lim m ( k = 1 m ( ln k ) n k - ( ln m ) n + 1 n + 1 ) .
2: 25.6 Integer Arguments
25.6.12 ζ ′′ ( 0 ) = - 1 2 ( ln ( 2 π ) ) 2 + 1 2 γ 2 - 1 24 π 2 + γ 1 ,
where γ 1 is given by (25.2.5). …
3: Errata
  • Subsection 25.2(ii) Other Infinite Series

    It is now mentioned that (25.2.5), defines the Stieltjes constants γ n . Consequently, γ n in (25.2.4), (25.6.12) are now identified as the Stieltjes constants.

  • 4: 31.15 Stieltjes Polynomials
    §31.15 Stieltjes Polynomials
    §31.15(ii) Zeros
    This is the Stieltjes electrostatic interpretation. …
    §31.15(iii) Products of Stieltjes Polynomials
    5: 1.14 Integral Transforms
    §1.14(vi) Stieltjes Transform
    The Stieltjes transform of a real-valued function f ( t ) is defined by … …
    Inversion
    Laplace Transform
    6: 9.10 Integrals
    9.10.15 0 e - p t Ai ( - t ) d t = 1 3 e p 3 / 3 ( Γ ( 1 3 , 1 3 p 3 ) Γ ( 1 3 ) + Γ ( 2 3 , 1 3 p 3 ) Γ ( 2 3 ) ) , p > 0 ,
    9.10.16 0 e - p t Bi ( - t ) d t = 1 3 e p 3 / 3 ( Γ ( 2 3 , 1 3 p 3 ) Γ ( 2 3 ) - Γ ( 1 3 , 1 3 p 3 ) Γ ( 1 3 ) ) , p > 0 .
    For the confluent hypergeometric function F 1 1 and the incomplete gamma function Γ see §§13.1, 13.2, and 8.2(i). …
    9.10.17 0 t α - 1 Ai ( t ) d t = Γ ( α ) 3 ( α + 2 ) / 3 Γ ( 1 3 α + 2 3 ) , α > 0 .
    §9.10(vii) Stieltjes Transforms
    7: 18.1 Notation
  • Racah: R n ( x ; α , β , γ , δ ) .

  • Dual Hahn: R n ( x ; γ , δ , N ) .

  • Stieltjes–Wigert: S n ( x ; q ) .

  • q -Racah: R n ( x ; α , β , γ , δ | q ) .

  • Triangle: P m , n α , β , γ ( x , y ) .

  • 8: 2.6 Distributional Methods
    §2.6(ii) Stieltjes Transform
    The Stieltjes transform of f ( t ) is defined by … For a more detailed discussion of the derivation of asymptotic expansions of Stieltjes transforms by the distribution method, see McClure and Wong (1978) and Wong (1989, Chapter 6). Corresponding results for the generalized Stieltjes transform …An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). …
    9: Bibliography C
  • M. Carmignani and A. Tortorici Macaluso (1985) Calcolo delle funzioni speciali Γ ( x ) , log Γ ( x ) , β ( x , y ) , erf ( x ) , erfc ( x ) alle alte precisioni. Atti Accad. Sci. Lett. Arti Palermo Ser. (5) 2(1981/82) (1), pp. 7–25 (Italian).
  • B. W. Char (1980) On Stieltjes’ continued fraction for the gamma function. Math. Comp. 34 (150), pp. 547–551.
  • R. F. Christy and I. Duck (1961) γ rays from an extranuclear direct capture process. Nuclear Physics 24 (1), pp. 89–101.
  • T. Clausen (1828) Über die Fälle, wenn die Reihe von der Form y = 1 + α 1 β γ x + α α + 1 1 2 β β + 1 γ γ + 1 x 2 + etc. ein Quadrat von der Form z = 1 + α 1 β γ δ ϵ x + α α + 1 1 2 β β + 1 γ γ + 1 δ δ + 1 ϵ ϵ + 1 x 2 + etc. hat. J. Reine Angew. Math. 3, pp. 89–91.