The Lambert W Function on the Reals

Manuel Eberl

24 April 2020


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.

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