Skip to main content

For questions related to the Lambert W or product log function, the inverse of $f(z)=ze^z$.

Equations involving exponentials multiplied to linear terms, such as $x+2=e^x$, cannot be solved using elementary functions in general. However, they may be solved in closed form using the Lambert W function, the inverse of $f(z)=ze^z$. In the given example: $$x+2=e^x$$ $$(-x-2)e^{-x-2}=-e^{-2}$$ $$-x-2=W(-e^{-2})$$ $$x=-W(-e^{-2})-2$$ There are two real branches of the Lambert W: $$W_0(z)\text{ for }z\in[-1/e,\infty)$$ $$W_{-1}(z)\text{ for }z\in[-1/e,0)$$

There are also branches $W_n(z)$, with $n\in\mathbb{Z}$, which are defined only on the complex plane, but which still respect the property of solving $W_n(z)e^{W_n(z)}=z$.

The regions of the complex plane where the inverse relationship occurs are bounded by the curve $x=-y\cot(y)$.

As discussed in the Corless et al. reference, the use of $W$ follows early Maple usage. Exact solutions to some mathematical models in the natural sciences, such as Michaelis–Menten kinetics, use this function.