The Principal Branch of the Lambert W Function

Abstract

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