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relation to polylogarithms


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1: 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: …
2: 25.12 Polylogarithms
The special case z = 1 is the Riemann zeta function: ζ ( s ) = Li s ( 1 ) . … Further properties include …and … In terms of polylogarithms
3: 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 , ,
4: 25.1 Special Notation
k , m , n nonnegative integers.
primes on function symbols: derivatives with respect to argument.
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 ) .
5: 25.18 Methods of Computation
The principal tools for computing ζ ( s ) are the expansion (25.2.9) for general values of s , and the Riemann–Siegel formula (25.10.3) (extended to higher terms) for ζ ( 1 2 + i t ) . …Calculations relating to derivatives of ζ ( s ) and/or ζ ( s , a ) can be found in Apostol (1985a), Choudhury (1995), Miller and Adamchik (1998), and Yeremin et al. (1988). 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). … Calculations to date (2008) have found no nontrivial zeros off the critical line. …
6: Software Index
  • Research Software.

    This is software of narrow scope developed as a byproduct of a research project and subsequently made available at no cost to the public. The software is often meant to demonstrate new numerical methods or software engineering strategies which were the subject of a research project. When developed, the software typically contains capabilities unavailable elsewhere. While the software may be quite capable, it is typically not professionally packaged and its use may require some expertise. The software is typically provided as source code or via a web-based service, and no support is provided.

  • Open Source Collections and Systems.

    These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

  • Software Associated with Books.

    An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.

  • Commercial Software.

    Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

  • Guide to Available Mathematical Software

    A cross index of mathematical software in use at NIST.

  • 7: Bibliography L
  • S. Lewanowicz (1985) Recurrence relations for hypergeometric functions of unit argument. Math. Comp. 45 (172), pp. 521–535.
  • S. Lewanowicz (1987) Corrigenda: “Recurrence relations for hypergeometric functions of unit argument” [Math. Comp. 45 (1985), no. 172, 521–535; MR 86m:33004]. Math. Comp. 48 (178), pp. 853.
  • L. Lewin (1981) Polylogarithms and Associated Functions. North-Holland Publishing Co., New York.
  • L. Lorch and M. E. Muldoon (2008) Monotonic sequences related to zeros of Bessel functions. Numer. Algorithms 49 (1-4), pp. 221–233.
  • J. D. Louck (1958) New recursion relation for the Clebsch-Gordan coefficients. Phys. Rev. (2) 110 (4), pp. 815–816.
  • 8: Bibliography V
  • G. Valent (1986) An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.
  • O. Vallée and M. Soares (2010) Airy Functions and Applications to Physics. Second edition, Imperial College Press, London.
  • 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.
  • H. Volkmer (1982) Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.
  • 9: Bibliography K
  • A. Khare, A. Lakshminarayan, and U. Sukhatme (2003) Cyclic identities for Jacobi elliptic and related functions. J. Math. Phys. 44 (4), pp. 1822–1841.
  • S. Koizumi (1976) Theta relations and projective normality of Abelian varieties. Amer. J. Math. 98 (4), pp. 865–889.
  • 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.
  • M. D. Kruskal (1974) The Korteweg-de Vries Equation and Related Evolution Equations. In Nonlinear Wave Motion (Proc. AMS-SIAM Summer Sem., Clarkson Coll. Tech., Potsdam, N.Y., 1972), A. C. Newell (Ed.), Lectures in Appl. Math., Vol. 15, pp. 61–83.