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I have this negative number $x<-1$ (thus in the lower branch of the Lambert W function). Is it possible to find $W_{0}(xe^x)$ in terms of $x$ in a useful/non-trivial form?

resgh
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1 Answers1

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By the definition of the Lambert W function, the value $y = W_0\left(x \mathrm{e}^{x}\right)$ satisfies: $$ y \mathrm{e}^{y} = x \mathrm{e}^{x} $$ There are two real solutions to this equation for each $x<-1$. One $y\leqslant -1$ and another $-1 \leqslant y < 0$. The solutions are $W_{-1}\left(x \mathrm{e}^{x}\right) = x$ and $W_0\left( x \mathrm{e}^{x} \right)$. The latter solution is a non-trivial function of $x$. enter image description here

Sasha
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  • I understand that the function is non-trivial on x. I meant to ask for some form of expression that is non-trivial/useful. A perfect example of what I think is considered trivial is $W_{0}(xe^x)$. I hope you understand my question better. – resgh Aug 30 '13 at 19:40