On the Lambert \(w\) function
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 \(w\).
scientific article; zbMATH DE number 947614
scientific article