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1: 27.7 Lambert Series as Generating Functions
§27.7 Lambert Series as Generating Functions
β–ΊLambert series have the form …
2: Bibliography S
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  • 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: Bibliography C
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  • R. M. Corless, D. J. Jeffrey, and D. E. Knuth (1997) A sequence of series for the Lambert W function. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (Kihei, HI), pp. 197–204.
  • 4: 4.13 Lambert W -Function
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    4.13.10 W k ⁑ ( z ) ξ k ln ⁑ ξ k + n = 1 ( 1 ) n ξ k n ⁒ m = 1 n [ n n m + 1 ] ⁒ ( ln ⁑ ξ k ) m m ! ,
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    4.13.11 W ± 1 ⁑ ( x βˆ“ 0 ⁒ i ) Ξ· ln ⁑ Ξ· + n = 1 1 Ξ· n ⁒ m = 1 n [ n n m + 1 ] ⁒ ( ln ⁑ Ξ· ) m m ! ,
    5: Errata
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  • Expansion

    §4.13 has been enlarged. The Lambert W -function is multi-valued and we use the notation W k ⁑ ( x ) , k β„€ , for the branches. The original two solutions are identified via Wp ⁑ ( x ) = W 0 ⁑ ( x ) and Wm ⁑ ( x ) = W ± 1 ⁑ ( x βˆ“ 0 ⁒ i ) .

    Other changes are the introduction of the Wright Ο‰ -function and tree T -function in (4.13.1_2) and (4.13.1_3), simplification formulas (4.13.3_1) and (4.13.3_2), explicit representation (4.13.4_1) for d n W d z n , additional Maclaurin series (4.13.5_1) and (4.13.5_2), an explicit expansion about the branch point at z = e 1 in (4.13.9_1), extending the number of terms in asymptotic expansions (4.13.10) and (4.13.11), and including several integrals and integral representations for Lambert W -functions in the end of the section.

  • 6: 4.45 Methods of Computation
    β–ΊThe function ln ⁑ x can always be computed from its ascending power series after preliminary scaling. … β–ΊThe function arctan ⁑ x can always be computed from its ascending power series after preliminary transformations to reduce the size of x . … β–Ί
    §4.45(iii) Lambert W -Function
    β–ΊFor 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)). … β–ΊSimilarly for Wm ⁑ ( x ) in the interval [ 1 / e , 0 ) . …
    7: Bibliography J
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  • D. Jacobs and F. Lambert (1972) On the numerical calculation of polylogarithms. Nordisk Tidskr. Informationsbehandling (BIT) 12 (4), pp. 581–585.
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  • 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.
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  • D. J. Jeffrey and N. Murdoch (2017) Stirling Numbers, Lambert W and the Gamma Function. In Mathematical Aspects of Computer and Information Sciences, J. Blömer, I. S. Kotsireas, T. Kutsia, and D. E. Simos (Eds.), Cham, pp. 275–279.
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  • 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.
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  • 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.
  • 8: Bibliography K
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  • G. A. Kalugin and D. J. Jeffrey (2011) Unimodal sequences show that Lambert W is Bernstein. C. R. Math. Acad. Sci. Soc. R. Can. 33 (2), pp. 50–56.
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  • G. A. Kalugin, D. J. Jeffrey, and R. M. Corless (2012) Bernstein, Pick, Poisson and related integral expressions for Lambert W . Integral Transforms Spec. Funct. 23 (11), pp. 817–829.
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  • 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.
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  • A. I. Kheyfits (2004) Closed-form representations of the Lambert W function. Fract. Calc. Appl. Anal. 7 (2), pp. 177–190.
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  • 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.
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  • H. Maass (1971) Siegel’s modular forms and Dirichlet series. Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.
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  • I. MezΕ‘ (2020) An integral representation for the Lambert W function.
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  • G. F. Miller (1966) On the convergence of the Chebyshev series for functions possessing a singularity in the range of representation. SIAM J. Numer. Anal. 3 (3), pp. 390–409.
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  • L. J. Mordell (1958) On the evaluation of some multiple series. J. London Math. Soc. (2) 33, pp. 368–371.
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  • C. Mortici (2011a) A new Stirling series as continued fraction. Numer. Algorithms 56 (1), pp. 17–26.