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What is a good approximation for the Lambert $W(x)$ function for values between $\frac{-1}{e}$ and $0$? Is it simply $x-x^2$? If so, what bounds are there on the error?

vonbrand
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1 Answers1

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See

  • R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, D. E. Knuth. On the Lambert W Function, Advances in Computational Mathematics 5(1), 1996, 329–359. doi:10.1007/BF02124750 (preprint)

for pretty much everything you might want to know about the Lambert $W$ function.

In particular, you probably want formula (4.22) which expresses the principal branch of the $W$ function as $$ W(z) = -1 + p - p^2/3 + (11/72)p^3 + \dots, $$ where $p = \sqrt{2(ez+1)}$. Formulas (4.23) and (4.24) define the coefficients of this series.

(If you are interested in the branch taking values below $-1$, then (4.20) is a good place to start. But your reference to $x - x^2$ implies that you are interested in the principal branch.)

András Salamon
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