Revision as of 05:23, 16 September 2016

The Lambert $W$ function is the (multi-valued) inverse of the function $f(x)=xe^{x}$.

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-The Lambert $W$ function is the (multi-valued) inverse of the function $g(x)=xe^{x}$. The function $g$ is not [[injective]] because [http://www.wolframalpha.com/input/?i=plot+y%3Dxe^x+for+-1%3Cy%3C2 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.
+The Lambert $W$ function is the (multi-valued) inverse of the function $f(x)=xe^{x}$.
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-File:lambertwplot.png|Graph of branches $W_0(x)$ and $W_1(x)$ on $[-1,1]$.
+File:Lambertw0plot.png|Plot of the principal branch $W_0$.$.
 File:Complexlambertw0.png|[[Domain coloring]] of [[analytic continuation]] of branch $W_0(x)$ to $\mathbb{C}$.
 File:Complexlambertw-1.png|[[Domain coloring]] of [[analytic continuation]] of branch $W_{-1}(x)$ to $\mathbb{C}$.