# Lambert series

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## 8 matching pages

##### 1: 27.7 Lambert Series as Generating Functions
###### §27.7 LambertSeries 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 $W\left(x\right)$ is the solution of the equation … We call the solution for which $W\left(x\right)\geq W\left(-1/e\right)$ the principal branch and denote it by $\mathrm{Wp}\left(x\right)$. … 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 $\operatorname{arctan}x$ 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 $x\in[-1/e,\infty)$ the principal branch $\mathrm{Wp}\left(x\right)$ can be computed by solving the defining equation $We^{W}=x$ numerically, for example, by Newton’s rule (§3.8(ii)). … Similarly for $\mathrm{Wm}\left(x\right)$ 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 $\nu$ is real and $0\leq\mu<\frac{1}{2}$.” It has been corrected to read “If $0\leq\Re\mu<\frac{1}{2}$ then they converge, but, if $\theta\not=\frac{1}{2}\pi$, they do not converge absolutely.”

Reported by Hans Volkmer on 2021-06-04

• Equation (4.13.11)
4.13.11 $\mathrm{Wm}\left(x\right)=-\eta-\ln\eta-\frac{\ln\eta}{\eta}+\frac{(\ln\eta)^{% 2}}{2\eta^{2}}-\frac{\ln\eta}{\eta^{2}}+O\left(\frac{(\ln\eta)^{3}}{\eta^{3}}\right)$

Originally the sign in front of $\frac{(\ln\eta)^{2}}{2\eta^{2}}$ was $-$. The correct sign is $+$.

• Subsection 25.2(ii) Other Infinite Series

It is now mentioned that (25.2.5), defines the Stieltjes constants $\gamma_{n}$. Consequently, $\gamma_{n}$ 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, $\mathrm{Wm}\left(n\right)$ was replaced by $\mathrm{Wp}\left(n\right)$ twice, and the wording was changed from “or, equivalently, $N={\mathrm{e}}^{\mathrm{Wm}\left(n\right)}$” to “or, specifically, $N={\mathrm{e}}^{\mathrm{Wp}\left(n\right)}$”.

Reported by Gergő Nemes on 2018-04-09