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On the Lambert $w$ function. (English) Zbl 0863.65008
The tree function $T$ defined by series $T\left(v\right)=v+\frac{2}{2!}{v}^{2}+\frac{{3}^{2}}{3!}{v}^{3}+\frac{{4}^{3}}{4!}{v}^{4}+\cdots$ converges for $|v|<\frac{1}{e}$. It equals $-w\left(-v\right)$, where $w\left(z\right)$ is defined to be the function satisfying $w\left(z\right){e}^{w\left(z\right)}=z$. 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 $w$.

MSC:
 65E05 Numerical methods in complex analysis 30E10 Approximation in the complex domain 65D20 Computation of special functions, construction of tables
Maple
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