The Lambert W function

The Lambert W function, named after Johann Heinrich Lambert (1728 to 1777), is specified by:

W(z)e^{W(z)}=z

W is multivalued, even when z and W are real. W(x) has a single value when x \ge 0, x \in \mathbb{R}. That is referred to as the principal branch and denoted W_0.

There are two values of W when -1/e \lt x \lt 0. One value is on the principal branch and W_0(x) \ge -1. The other value is on a branch denoted W_{-1} and W_{-1}(x) \le -1.

As:

(-1)e^{-1}=-1/e

W(-1/e)=-1. That point is considered to be on both the principal branch and W_{-1}.

You can see that there are two values by identifying that there is a single maximum or minimum of x=We^W:

\frac {d}{dW} (We^W) = e^W+We^W=e^W(1+W)=0 \Rightarrow W=-1

and that:

\lim_{W \to -\infty} x = 0^-, \quad \lim_{W \to \infty} x = \infty