Lambert W of Zero is Zero

Theorem

Let $W_0$ denote principal branch of the Lambert W function.

Then:

$W_0 \left({0}\right) = 0$

Proof

From the definition of the principal branch of the Lambert W function:

$y = W_0 \left({x}\right) \iff x = y e^y$

where $x \in \left[{-\dfrac 1 e \,.\,.\, \to}\right)$ and $y \in \left[{-1 \,.\,.\, \to}\right)$.

The result follows from substituting $x = 0$ and $y = 0$.

$\blacksquare$