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Zbl 0863.65008
Corless, R.M.; Gonnet, G.H.; Hare, D.E.G.; Jeffrey, D.J.; Knuth, D.E.
On the Lambert $w$ function. (English)
[J] Adv. Comput. Math. 5, No.4, 329-359 (1996). ISSN 1019-7168; ISSN 1572-9044/e

The tree function $T$ defined by series $T(v)=v+{2\over 2!}v^2+{3^2\over 3!}v^3+{4^3\over 4!}v^4+\dots$ converges for $|v|< {1\over e}$. It equals $-w(-v)$, where $w(z)$ is defined to be the function satisfying $w(z)e^{w(z)}=z$. This paper discusses both $w$ and $T$, concentrating on $w$. The authors present a new discussion of the complex branches for $w$, an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containing $w$.
[R.S.Dahiya (Ames)]

MSC 2000:: *65E05 Numerical methods in complex analysis
30E10 Approximation in the complex domain
65D20 Computation of special functions

Keywords: Lambert $w$ function; tree function; asymptotic expansion; symbolic integration

Cited in: Zbl 0999.26002

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