### Abstract

The Lambert *W* function is a multi-valued
function defined as the inverse function of *x*
↦ *x*
e^{x}. Besides numerous
applications in combinatorics, physics, and engineering, it also
frequently occurs when solving equations containing both
e^{x} and
*x*, or both *x* and log
*x*.

This article provides a
definition of the two real-valued branches
*W*_{0}(*x*)
and
*W*_{-1}(*x*)
and proves various properties such as basic identities and
inequalities, monotonicity, differentiability, asymptotic expansions,
and the MacLaurin series of
*W*_{0}(*x*)
at *x* = 0.