Explicit and recursive estimates of the Lambert W function
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Solutions to a wide variety of transcendental equations can be expressed in terms of the Lambert W function. The W function, occurring frequently in applications, is a non-elementary, but now standard mathematical function implemented in all major technical computing systems. In this work, we discuss some approximations of the two real branches, W_0 and W_-1. On the one hand, we present some analytic lower and upper bounds on W_0 for large arguments that improve on some earlier results in the literature. On the other hand, we analyze two logarithmic recursions, one with linear, and the other with quadratic rate of convergence. We propose suitable starting values for the recursion with quadratic rate that ensure convergence on the whole domain of definition of both real branches. We also provide a priori, simple, explicit and uniform estimates on its convergence speed that enable guaranteed, high-precision approximations of W_0 and W_-1 at any point. Finally, as an application of the W_0 function, we settle a conjecture about the growth rate of the positive non-trivial solutions to the equation x^y=y^x.
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