The Lambert's Omega function has 2 real branches denoted by $W_{-1}(x)$ and $W_0(x)$ and it represents the solution(s) of the equation $xe^x=a$.

I learned that this function can be generalized and for every $n\in \Bbb R$ so that $W_n(x)$ in the solutions to the equation $xe^x+nx=a$.

My problem is this: if I set $n=0$ I get the original equation so everything makes sense to me but if I set $n=-1$ do I get again my original definition of the branch $W_{-1}(x)$ ?

My equation becomes this: $xe^x-x=a$

Is it solvable with the "classical" Lambert's function (the negative branch of it) ? Probably I'm missing something very trivial but I can't see how to come back to the original definition of $W(x)$ this time.

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    $\begingroup$ Both $W_0(x)$ and $W_{-1}(x)$ are parts of the complete solution of the equation $We^W=x$. The index $0$ or $-1$ do not refer to a more general equation. This is only a conventional symbol to distinguish the two branches. So, there is no relationship with the parameter $n$ of the equation $We^W+nW=x$. This is confusing. It should be better to write $W(n,x)$ instead of $W_n(x)$. And in case of two branches $W_0(n,x)$ and $W_{-1}(n,x)$ $\endgroup$ – JJacquelin May 13 '15 at 14:33
  • $\begingroup$ Okay now I got this: every n-Lambert function has 1 or 2 branches that we shall denote $W_{n;-1}$ and $W_{n;0}$ and this turns out to be the "classical" Lambert's Omega function for $n=0$. If you post your comment as an answer I'll accept it. $\endgroup$ – Renato Faraone May 13 '15 at 15:09
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    $\begingroup$ If the the equation is even more generalized to real $\nu$ instead of integer $n$, $$We^W+\nu W=x$$ then, for some range of $\nu$, they are three branches. I Wonder what would be the symbol ! $\endgroup$ – JJacquelin May 13 '15 at 15:42
  • $\begingroup$ The 3 branches could be $W_{n;-1}$, $W_{n;0}$ and $W_{n;1}$. I'm wondering how many branches there could be if we allow $n$ to be complex, we could have more branches for every branch of the function in the complex plane! $\endgroup$ – Renato Faraone May 13 '15 at 15:49
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    $\begingroup$ Of course, the Lambert function is already a standard function referenced in the handbooks of special functions and implemented in most of the softwares. It is not the same for it's various generalizations. Don't mention my help, it was a pleasure. $\endgroup$ – JJacquelin May 13 '15 at 17:02

This is a graphical answer to the question :

enter image description here

I don't know if there is a standardized symbol for this sort of generalized Lambert-W function. The symbols used above might be non-standard.


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