The Lambert W-function is the solution to the transcendental equation W(x)eW(x) = x. It has two real branches, one of which, for x ∈ [−1/e, ∞], is usually denoted as the principal branch. On this branch, the function grows from − 1 to infinity, logarithmically at large x. The present work is devoted to the construction of accurate approximations for the principal branch of the W-function. In particular, a simple, global analytic approximation is derived that covers the whole branch with a maximum relative error smaller than 5 × 10−3. Starting from it, machine precision accuracy is reached everywhere with only three steps of a quadratically convergent iterative scheme, here examined for the first time, which is more efficient than standard Newton’s iteration at large x. Analytic bounds for W are also constructed, for x > e, which are much tighter than those currently available. It is noted that the exponential of the upper bounding function yields an upper bound for the prime counting function π(n) that is better than the well-known Chebyshev’s estimates at large n. Finally, the construction of accurate approximations to W based on Chebyshev spectral theory is discussed; the difficulties involved are highlighted, and methods to overcome them are presented. © 2017 Springer Science+Business Media New York

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