Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

There are well known bounds for $W_0$, the "principal" real-valued branch of the Lambert-W function. For example, $W_0(x)$ lies between $\log x - \log\log x$ and $\log x - \frac{1}{2}\log \log x$, and associated big-O asymptotics of $W_0$ as $x\rightarrow\infty$.

What about bounds for $W_{-1}(x)$, the other real-valued branch of the Lambert-W function as $x\rightarrow 0^-$? The limit is well-known to be $-\infty$, but how fast does $W_{-1}(x)$ approach the limit? Are $O$ asymptotics known in terms of more elementary functions such as powers and logs?

share|improve this question

1 Answer 1

up vote 1 down vote accepted

First, if $x \in ]-1/e,0[$, one has $1 < -W_{-1}(x) \leq -\frac{1}{x}$ (it is easy to prove that these inequalities are equivalent to $-1/e < x \leq \frac{\exp(1/x)}{x}$, and that is true here). Then $0 < \ln(-W_{-1}(x)) \leq -\ln(-x)$.

Since $W_{-1}(x)\exp(W_{-1}(x)) = x$, the above inequalities yield $1 < -W_{-1}(x) = -\ln(-x)+\ln(-W_{-1}(x)) \leq -2\ln(-x)$ (of course, one cas replace 2 by another suitable constant, if I only look for an asymptotic behavior, as we shall see). Finally, this gives $0 < \ln(-W_{-1}(x)) < 2\ln(-\ln(-x))$. Combine everything here, and you already have $W_{-1}(x) = \ln(-x) + O(\ln(-\ln(-x)))$ as $x \to 0^-$. One can do better, by reinjecting this estimate in the identity $W_{-1}(x) = \ln(-x) - \ln(-W_{-1}(x))$, and one obstains $W_{-1}(x) = \ln(-x) - \ln(-\ln(-x)) + O\left(\frac{\ln(-\ln(-x))}{\ln(-x)}\right)$.

You may find some interets in reading "Asymptotic methods in analysis" by De Bruijn, the methods used to write $W_0$ as an infinite sum of powers involving $\ln$ can be used with $W_{-1}$, I guess.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.