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1: 25.12 Polylogarithms
§25.12 Polylogarithms
§25.12(ii) Polylogarithms
For real or complex s and z the polylogarithm Li s ( z ) is defined by …
Integral Representation
In terms of polylogarithms
2: 25.1 Special Notation
The main related functions are the Hurwitz zeta function ζ ( s , a ) , the dilogarithm Li 2 ( z ) , the polylogarithm Li s ( z ) (also known as Jonquière’s function ϕ ( z , s ) ), Lerch’s transcendent Φ ( z , s , a ) , and the Dirichlet L -functions L ( s , χ ) .
3: 25.13 Periodic Zeta Function
§25.13 Periodic Zeta Function
The notation F ( x , s ) is used for the polylogarithm Li s ( e 2 π i x ) with x real: …
4: 25.18 Methods of Computation
For the Hurwitz zeta function ζ ( s , a ) see Spanier and Oldham (1987, p. 653) and Coffey (2009). For dilogarithms and polylogarithms see Jacobs and Lambert (1972), Osácar et al. (1995), Spanier and Oldham (1987, pp. 231–232), and Zudilin (2007). …
5: 25.19 Tables
  • 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.

  • 6: 25.14 Lerch’s Transcendent
    The Hurwitz zeta function ζ ( s , a ) 25.11) and the polylogarithm Li s ( z ) 25.12(ii)) are special cases:
    25.14.2 ζ ( s , a ) = Φ ( 1 , s , a ) , s > 1 , a 0 , - 1 , - 2 , ,
    25.14.3 Li s ( z ) = z Φ ( z , s , 1 ) , s > 1 , | z | 1 .
    7: 25.21 Software
    §25.21(v) Dilogarithms, Polylogarithms
    8: Bibliography J
  • D. Jacobs and F. Lambert (1972) On the numerical calculation of polylogarithms. Nordisk Tidskr. Informationsbehandling (BIT) 12 (4), pp. 581–585.
  • 9: Bibliography V
  • A. J. van der Poorten (1980) Some Wonderful Formulas an Introduction to Polylogarithms. In Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979), R. Ribenboim (Ed.), Queen’s Papers in Pure and Appl. Math., Vol. 54, Kingston, Ont., pp. 269–286.
  • 10: Bibliography K
  • K. S. Kölbig, J. A. Mignaco, and E. Remiddi (1970) On Nielsen’s generalized polylogarithms and their numerical calculation. Nordisk Tidskr. Informationsbehandling (BIT) 10, pp. 38–73.
  • K. S. Kölbig (1986) Nielsen’s generalized polylogarithms. SIAM J. Math. Anal. 17 (5), pp. 1232–1258.