# Thread: The product log is not elementary

1. ## The product log is not elementary

The product logarithm (Lambert W) is the inverse of $\displaystyle f(z) = z e^z$. Wikipedia claims it is not elementary, but gives no reference. How do we know that is not elementary?

2. Originally Posted by maddas
The product logarithm (Lambert W) is the inverse of $\displaystyle f(z) = z e^z$. Wikipedia claims it is not elementary, but gives no reference. How do we know that is not elementary?

You can read a proof here: Algebraic properties of the Lambert W function from a result of Rosenlicht and of Liouville - Integral Transforms and Special Functions

It is not trivial, or even easy.

Tonio