Definition:Lambert W Function/Principal Branch/Real Valued

Definition
The principal branch of the Lambert W function is the real function $W_0: \left[{-\dfrac 1 e \,.\,.\, \to}\right) \to \left[{-1 \,.\,.\, \to}\right)$ such that:
 * $x = W_0 \left({x}\right) e^{W_0 \left({x}\right)}$

Also denoted as
When the principal branch is the only branch under consideration, the subscript is sometimes omitted:


 * $x = W \left({x}\right) e^{W \left({x}\right)}$