Bounds on the Lambert Function and Their Application to the Outage Analysis of User Cooperation

Abstract

Problems formulated in terms of logarithmic or exponential equations often use the Lambert W function in their solutions. Expansions, approximations and bounds on W have been derived in an effort to gain a better understanding of the relationship between equation parameters. In this paper, we focus on one of the branches of W, denoted as W <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sub> , we derive tractable upper and lower bounds and we illustrate their usefulness in identifying conditions under which user cooperation can yield a lower outage probability than non-cooperative transmission.