TITLE: Inverting a function: proof of $W(x) = \ln\frac{x}{\ln\frac{x}{\ddots}}$
QUESTION [5 upvotes]: for $|W(x)|>1$, $W$ The Lambert W-function:
$$ W(x) =\ln\cfrac{x}{\ln\cfrac{x}{\ln\cfrac{x}{\ddots}}}  $$

REPLY [2 votes]: A simple proof by recursive definition:
$$
f(x)=\ln\cfrac{x}{\ln\cfrac{x}{\ln\cfrac{x}{\ddots}}}\\
f(x)=\ln\frac{x}{f(x)}\\
e^{f(x)}=\frac{x}{f(x)}\\
f(x)e^{f(x)}=x\\
f(x)=W(x)
$$