# Lambert W function

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

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

for all $x\in\mathbbmss{C}$. 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’ page http://kong.apmaths.uwo.ca/ rcorless/frames/PAPERS/LambertW/“On the Lambert W Function

Title Lambert W function LambertWFunction 2013-03-22 12:40:48 2013-03-22 12:40:48 drini (3) drini (3) 8 drini (3) Definition msc 33B30 product log