The Lambert W Function on the Reals

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 W₀(x) and W_-1(x) and proves various properties such as basic identities and inequalities, monotonicity, differentiability, asymptotic expansions, and the MacLaurin series of W₀(x) at x = 0.