Lambert W function

Summary

The Lambert W function is the inverse of the map $w \mapsto w e^w$, so $W(x)\, e^{W(x)} = x$. Over the complex numbers it is multivalued, with countably many branches $W_k$; on the reals it has two branches $W_0$ and $W_{-1}$. It solves many equations that mix exponentials and polynomials, such as $x = e^{-x}$ or $a^x = x$. It appears in combinatorics, in delay differential equations and in physics.