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1: 27.7 Lambert Series as Generating Functions
§27.7 Lambert Series as Generating Functions
Lambert series have the form …
2: Bibliography S
  • T. C. Scott, G. Fee, and J. Grotendorst (2013) Asymptotic series of generalized Lambert W function. ACM Commun. Comput. Algebra 47 (3), pp. 75–83.
  • 3: 4.13 Lambert W -Function
    §4.13 Lambert W -Function
    The Lambert W -function W ( x ) is the solution of the equation … We call the solution for which W ( x ) W ( - 1 / e ) the principal branch and denote it by Wp ( x ) . … Properties include: … For integral representations of all branches of the Lambert W -function see Kheyfits (2004). …
    4: 4.45 Methods of Computation
    The function ln x can always be computed from its ascending power series after preliminary scaling. … The function arctan x can always be computed from its ascending power series after preliminary transformations to reduce the size of x . …
    §4.45(iii) Lambert W -Function
    For x [ - 1 / e , ) the principal branch Wp ( x ) can be computed by solving the defining equation W e W = x numerically, for example, by Newton’s rule (§3.8(ii)). … Similarly for Wm ( x ) in the interval [ - 1 / e , 0 ) . …
    5: Bibliography C
  • B. C. Carlson (2008) Power series for inverse Jacobian elliptic functions. Math. Comp. 77 (263), pp. 1615–1621.
  • H. S. Carslaw (1930) Introduction to the Theory of Fourier’s Series and Integrals. 3rd edition, Macmillan, London.
  • F. Chapeau-Blondeau and A. Monir (2002) Numerical evaluation of the Lambert W function and application to generation of generalized Gaussian noise with exponent 1/2. IEEE Trans. Signal Process. 50 (9), pp. 2160–2165.
  • C. W. Clenshaw (1955) A note on the summation of Chebyshev series. Math. Tables Aids Comput. 9 (51), pp. 118–120.
  • R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth (1996) On the Lambert W function. Adv. Comput. Math. 5 (4), pp. 329–359.
  • 6: Bibliography K
  • D. Karp, A. Savenkova, and S. M. Sitnik (2007) Series expansions for the third incomplete elliptic integral via partial fraction decompositions. J. Comput. Appl. Math. 207 (2), pp. 331–337.
  • M. Katsurada (2003) Asymptotic expansions of certain q -series and a formula of Ramanujan for specific values of the Riemann zeta function. Acta Arith. 107 (3), pp. 269–298.
  • A. I. Kheyfits (2004) Closed-form representations of the Lambert W function. Fract. Calc. Appl. Anal. 7 (2), pp. 177–190.
  • E. J. Konopinski (1981) Electromagnetic Fields and Relativistic Particles. International Series in Pure and Applied Physics, McGraw-Hill Book Co., New York.
  • C. Krattenthaler (1993) HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively q -binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.
  • 7: Bibliography J
  • D. Jacobs and F. Lambert (1972) On the numerical calculation of polylogarithms. Nordisk Tidskr. Informationsbehandling (BIT) 12 (4), pp. 581–585.
  • A. J. E. M. Janssen (2021) Bounds on Dawson’s integral occurring in the analysis of a line distribution network for electric vehicles. Eurandom Preprint Series Technical Report 14, Eurandom, Eindhoven, The Netherlands.
  • D. S. Jones, M. J. Plank, and B. D. Sleeman (2010) Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical and Computational Biology Series, CRC Press, Boca Raton, FL.
  • D. S. Jones and B. D. Sleeman (2003) Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL.
  • D. S. Jones (1964) The Theory of Electromagnetism. International Series of Monographs on Pure and Applied Mathematics, Vol. 47. A Pergamon Press Book, The Macmillan Co., New York.
  • 8: Errata
  • Equations (14.13.1), (14.13.2)

    Originally it was stated that these Fourier series converge “…conditionally when ν is real and 0 μ < 1 2 .” It has been corrected to read “If 0 μ < 1 2 then they converge, but, if θ 1 2 π , they do not converge absolutely.”

    Reported by Hans Volkmer on 2021-06-04

  • Equation (4.13.11)
    4.13.11 Wm ( x ) = - η - ln η - ln η η + ( ln η ) 2 2 η 2 - ln η η 2 + O ( ( ln η ) 3 η 3 )

    Originally the sign in front of ( ln η ) 2 2 η 2 was - . The correct sign is + .

  • 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.

  • Subsection 26.7(iv)

    In the final line of this subsection, Wm ( n ) was replaced by Wp ( n ) twice, and the wording was changed from “or, equivalently, N = e Wm ( n ) ” to “or, specifically, N = e Wp ( n ) ”.

    Reported by Gergő Nemes on 2018-04-09