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On the Lambert \(w\) function. (English) Zbl 0863.65008

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 \(w\).
Reviewer: R.S.Dahiya (Ames)

MSC:

65E05 General theory of numerical methods in complex analysis (potential theory, etc.)
30E10 Approximation in the complex plane
65D20 Computation of special functions and constants, construction of tables
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