Analytical approximations for real values of the Lambert W-function

The Lambert W is a transcendental function defined by solutions of the equation Wexp(W)=x. For real values of the argument, x, the W-function has two branches, W0 (the principal branch) and W−1 (the negative branch). A survey of the literature reveals that, in the case of the principal branch (W0), the vast majority of W-function applications use, at any given time, only a portion of the branch viz. the parts defined by the ranges −1≤W0≤0 and 0≤W0. Approximations are presented for each portion of W0, and for W−1. It is shown that the present approximations are very accurate with relative errors down to around 0.02% or smaller. The approximations can be used directly, or as starting values for iterative improvement schemes.