LambertWFunction 2002-05-28 00:03:11 2005-02-28 16:50:57 Definition \usepackage{graphicx} %\usepackage{xypic} \usepackage{bbm} \newcommand{\Z}{\mathbbmss{Z}} \newcommand{\C}{\mathbbmss{C}} \newcommand{\R}{\mathbbmss{R}} \newcommand{\Q}{\mathbbmss{Q}} \newcommand{\mathbb}[1]{\mathbbmss{#1}} \newcommand{\figura}[1]{\begin{center}\includegraphics{#1}\end{center}} \newcommand{\figuraex}[2]{\begin{center}\includegraphics[#2]{#1}\end{center}} Lambert's $W$ function is the inverse of the function $f: \C \to \C$ given by $f(x) := x e^x$. That is, $W(x)$ is the complex valued function that satisfies \begin{displaymath}W(x) e^{W(x)} = x,\end{displaymath} for all $x \in \mathbb{C}$. In practice the definition of $W(x)$ requires a branch cut, which is usually taken along the negative real axis. Lambert's W function is sometimes also called product log function. This function allow us to solve the functional equation $$g(x)^{g(x)}=x$$ since $$g(x)=e^{W(\ln(x))}.$$ \section{References} A site with good information on Lambert's W function is Corless' page \PMlinkexternal{``On the Lambert W Function''}{http://kong.apmaths.uwo.ca/~rcorless/frames/PAPERS/LambertW/}