Lambert W

The Lambert $W$ function is the (multi-valued) inverse of the function $g(x)=xe^{x}$. The function $g$ is not injective because its graph does not pass the "horizontal line test". Therefore the inverse function is multi-valued and not unique. This yields two branches of the $W$ function.

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  Graph of branches $W_0(x)$ and $W_1(x)$ on $[-1,1]$.

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  Domain coloring of analytic continuation of branch $W_0(x)$ to $\mathbb{C}$.

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  Domain coloring of analytic continuation of branch $W_{-1}(x)$ to $\mathbb{C}$.

References

Having fun with the Lambert $W(x)$ function

Videos

6: Recursion, Infinite Tetrations and the Lambert W Function