About the Project
NIST

Clausen integral

AdvancedHelp

(0.000 seconds)

6 matching pages

1: 25.19 Tables
  • Abramowitz and Stegun (1964) tabulates: ζ ( n ) , n = 2 , 3 , 4 , , 20D (p. 811); Li 2 ( 1 - x ) , x = 0 ( .01 ) 0.5 , 9D (p. 1005); f ( θ ) , θ = 15 ( 1 ) 30 ( 2 ) 90 ( 5 ) 180 , f ( θ ) + θ ln θ , θ = 0 ( 1 ) 15 , 6D (p. 1006). Here f ( θ ) denotes Clausen’s integral, given by the right-hand side of (25.12.9).

  • Cloutman (1989) tabulates Γ ( s + 1 ) F s ( x ) , where F s ( x ) is the Fermi–Dirac integral (25.12.14), for s = - 1 2 , 1 2 , 3 2 , 5 2 , x = - 5 ( .05 ) 25 , to 12S.

  • Fletcher et al. (1962, §22.1) lists many sources for earlier tables of ζ ( s ) for both real and complex s . §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of ζ ( s , a ) , and §22.17 lists tables for some Dirichlet L -functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.

  • 2: 25.21 Software
    §25.21(vi) Clausen’s Integral
    3: 25.12 Polylogarithms
    25.12.9 n = 1 sin ( n θ ) n 2 = - 0 θ ln ( 2 sin ( 1 2 x ) ) d x .
    The right-hand side is called Clausen’s integral. …
    4: Software Index
    5: Bibliography K
  • J. Kamimoto (1998) On an integral of Hardy and Littlewood. Kyushu J. Math. 52 (1), pp. 249–263.
  • E. L. Kaplan (1948) Auxiliary table for the incomplete elliptic integrals. J. Math. Physics 27, pp. 11–36.
  • N. M. Katz (1975) The congruences of Clausen-von Staudt and Kummer for Bernoulli-Hurwitz numbers. Math. Ann. 216 (1), pp. 1–4.
  • A. D. Kerr (1978) An indirect method for evaluating certain infinite integrals. Z. Angew. Math. Phys. 29 (3), pp. 380–386.
  • E. Kreyszig (1957) On the zeros of the Fresnel integrals. Canad. J. Math. 9, pp. 118–131.
  • 6: Bibliography C
  • L. Carlitz (1961b) The Staudt-Clausen theorem. Math. Mag. 34, pp. 131–146.
  • B. C. Carlson (1961b) Some series and bounds for incomplete elliptic integrals. J. Math. and Phys. 40, pp. 125–134.
  • B. C. Carlson (1965) On computing elliptic integrals and functions. J. Math. and Phys. 44, pp. 36–51.
  • 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.
  • D. Cvijović and J. Klinowski (1999) Integrals involving complete elliptic integrals. J. Comput. Appl. Math. 106 (1), pp. 169–175.