\begin{definition}[Definition:Lambert W Function/Principal Branch/Real Valued]
The principal branch of the Lambert W function is the real function $W_0: \hointr {-\dfrac 1 e} \to \to \hointr {-1} \to$ such that:
:$x = \map {W_0} x e^{\map {W_0} x}$
\end{definition}