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Let $y(x)=x\log(x)$ for $x>1$. Is it possible to write down the inverse function explicitly? Has this inverse function been named? (For example, the Bessel functions are "named" but cannot be defined explicitly using elementary functions.)

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The Lambert $W$ function is the inverse function of $g(x)=xe^x$, i.e. a function such that $W(x)\,e^{W(x)}=x$ for every $x$ in some range. To solve: $$ y \log y = x $$ by setting $y=e^{f(x)}$ is the same as solving $f(x) e^{f(x)}=x$, that gives $f(x)=W(x)$. It follows that: $$ y = e^{W(x)} = \frac{x}{W(x)}.$$

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  • $\begingroup$ Okay, this makes sense to me now - the crux is the substitution $y=e^{f(x)}$. Thanks. $\endgroup$ – FreshAir Oct 13 '15 at 1:52

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