The Lambert W Function on the Reals

Manuel Eberl 🌐

April 24, 2020

This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.


The Lambert W function is a multi-valued function defined as the inverse function of xx ex. Besides numerous applications in combinatorics, physics, and engineering, it also frequently occurs when solving equations containing both ex 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.


BSD License


Session Lambert_W