Parametric representation of the real branches $\operatorname{W_{0}},\operatorname{W_{-1}}$ of the Lambert W function

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$$\def\a{\alpha}\def\la{\ln{\a}}\def\e{\mathrm{e}}\def\W{\operatorname{W}}$$ $$\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}$$

For $$x\in(-\tfrac1{\mathrm{e}},0)$$, $$\a\in(0,1)$$ two real branches of the Lambert W function can be parameterized as follows:

\begin{align} \Wp(x)&= \frac{\a\ln\a}{1-\a} \tag{1}\label{1} ,\\ \Wm(x)&= \frac{\ln\a}{1-\a} \tag{2}\label{2} ,\\ x&=\Wp(x)\exp(\Wp(x)) =\Wm(x)\exp(\Wm(x)) \\ &= \a^{\tfrac a{1-\a}} \ln\Big(\a^{\tfrac a{1-\a}}\Big) = \a^{\tfrac1{1-\a}} \ln\Big(\a^{\tfrac1{1-\a}}\Big) . \end{align}

This parameterization is also consistent with the values of $$\Wp,\Wm$$ at $$x=0$$ and $$x=-\tfrac1{\mathrm{e}}$$: \begin{align} \Wp(0)&= \lim_{\a\to0}\frac{\a\ln\a}{1-\a}=0 ,\\ \Wp(-\tfrac1{\mathrm{e}})&= \lim_{\a\to1}\frac{\a\ln\a}{1-\a}=-1 ,\\ \Wm(0)&= \lim_{\a\to0}\frac{\ln\a}{1-\a}=-\infty ,\\ \Wm(-\tfrac1{\mathrm{e}})&= \lim_{\a\to1}\frac{\ln\a}{1-\a}=-1 . \end{align}

For any $$x\in(-\tfrac1{\mathrm{e}},0)$$ the value of corresponding parameter $$\a$$ is found as

\begin{align} \a(x)&=\frac{\Wp(x)}{\Wm(x)} . \end{align}

A dual form of \eqref{1}-\eqref{2} can be used for $$\a>1$$. In this case

\begin{align} \Wp(x)&= \frac{\ln\a}{1-\a} \tag{3}\label{3} ,\\ \Wm(x)&= \frac{\a\ln\a}{1-\a} \tag{4}\label{4} ,\\ \a(x)&=\frac{\Wm(x)}{\Wp(x)} . \end{align}

One related reference is

D. J. Jeffrey and J. E. Jankowski. "Branch Differences and Lambert W". In: 2014 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing. Sept. 2014, pp. 61-65. doi:10.1109/SYNASC.2014.16

where it is shown that the difference of branches of the Lambert $$\W$$ function is nontrivial.

Question: is there any other known references, where such parameterization is suggested/used?

Edit Also, in the fundamental paper

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J.Jeffrey, and D. E. Knuth. On the Lambert W Function. Advances in Computational Mathematics, 5:4 (1996) 329-359

there is a closely related example "Solution of a jet fuel problem", Eqns. 2.9-2.10, from which such parameterization can be easily deduced, but still, the parametric form of $$\Wp,\Wm$$ is not explicitly mentioned.

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• Have you looked at Corless, Gonnet, Hare, Jeffrey and Knuth? – Robert Israel May 20 at 15:25
• @Robert Israel: Yes, of course, this is a fundamental paper on the Lambert W function, and there is a related part (Solution of a jet fuel problem), but I'm looking for the reference, where an explicit parameterization of $\operatorname{W}_0$ and $\operatorname{W}_{-1}$ is stated/studied. – g.kov May 20 at 16:28