The Principal Branch of the Lambert W Function

Abstract

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