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The Lambert W function has two real branches: the principal branch and the secondary real branch: the former is denoted by $W_0$ or $W$, the latter by $W_{-1}$. How do we name them ?

For example, we can write: "Consider the solution $y = exp(-2 W(-x/2))$ where $W$ is the Lambert W function". In this case, the principal branch is implicitly assumed, so the issue is silently discarded. What can we write for the secondary case ? "Consider the solution $y = exp(-2 W_{-1}(-x/2))$ where $W_{-1}$ is ..."

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  • $\begingroup$ I think that upper branch for $W_0$ and lower branch for $W_{-1}$ are standard names. $\endgroup$ Commented Sep 24, 2015 at 4:31
  • $\begingroup$ "... where $W_{-1}$ is the lower branch Lambert W function" is correct ? $\endgroup$ Commented Sep 24, 2015 at 13:54
  • $\begingroup$ As you yourself have done, I too call $\mathrm{W}_{-1}(x)$ the secondary real branch. $\endgroup$
    – omegadot
    Commented Sep 24, 2015 at 17:32
  • $\begingroup$ Sorry, but I am still confused. Do you call $W_{-1}(x)$ the secondary real branch of the Lambert W function ? May I regard the branch itself as a function ? Let assume that the reader may not know a lot on (very) special functions. $\endgroup$ Commented Sep 25, 2015 at 2:45
  • $\begingroup$ Each branch where the Lambert W function is real is a function. Note that the inverse of the function $f(x) = x\mathrm{e}^x$ for all real $x$ is only a relation. Identifying the branch point between the two real branches occurs at $x=-1/\mathrm{e}$ leads to the two real branches for the Lambert W function - the principal branch $\mathrm{W}_0(x)$ and the secondary real branch $\mathrm{W}_{-1}(x)$ of the Lambert W function. $\endgroup$
    – omegadot
    Commented Sep 25, 2015 at 7:08

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