Lambert W function

Lambert’s $W$ function is the inverse of the function $f:ℂ\to ℂ$ given by $f(x):=x{e}^x$. That is, $W(x)$ is the complex valued function that satisfies

$$W(x)e^{W(x)}=x,$$

for all $x\in ℂ$. In practice the definition of $W(x)$ requires a branch cut, which is usually taken along the negative real axis. Lambert’s W function is sometimes also called product log function.

This function allow us to solve the functional equation

$$g{(x)}^{g(x)}=x$$

since

$$g(x)=e^{W(\ln (x))}.$$

1 References

A site with good information on Lambert’s W function is Corless’ pagehttp://kong.apmaths.uwo.ca/ rcorless/frames/PAPERS/LambertW/“On the Lambert W Function”