The Principal Branch of the Lambert W Function

Abstract

The Lambert W function is the multi-valued inverse of the function E(z)=zexpz$$E(z) = z \exp z$$. Let W~$$\widetilde{W}$$ be a branch of W defined and single-valued on a region D~$$\widetilde{D}$$. We show how to use the Taylor expansion of W~$$\widetilde{W}$$ at a given point of D~$$\widetilde{D}$$ to obtain an infinite series representation of W~$$\widetilde{W}$$ throughout D~$$\widetilde{D}$$.