Is there a compendium of equations that are solvable in terms of Lambert W?

Question

Such a compilation would list equations in the most general form possible along with their solutions using Lambert W, and a reference to the derivation. It would also mention equations for which no such solutions are possible.

Answer 1

As user1952009 notes, anything that reduces down to $ze^z=u$ is solvable for $z$ in terms of the Lambert W function, assuming $u$ is a constant to $z$.

There are a few general cases:

$$p^{ax+b}=cx+d\implies x=-\frac{W\left(-\frac{a\ln(p)}cp^{b-\frac{ad}c}\right)}{a\ln(p)}-\frac dc$$

$$x^x=z\implies x=e^{W(\ln(z))}$$

Now, anything of the form $f(x)e^{f(x)}=z$ has the solution $x=f^{-1}(W(z))$, which can be used to solve things like $xe^xe^{xe^x}=z$, which has a solution $x=W(W(z))$.

However, I will note that most equations that have $x$ in power towers where the difference in 'level' from the top $x$ to bottom $x$ is greater than $2$, is generally not solvable with the Lambert W function.

For example, $x^{x^x}=z$ is not solvable for $x$.

A few rarer examples that are solvable for $x$ often require more than 'algebraic' manipulations

$$x=e^x\implies x=e^x=e^{e^x}=e^{e^{e^x}}=\dots$$

We see that an original solution for $x$ on the left must also be a solution for $x$ on the right (but not the other way around). From this, we can at least find some of the solutions for $x$ in $x=e^{e^x}$.

So you could say problems of the form $x=f(f(f(\dots f(x)\dots)))$ have some solutions that can be found from $x=f(x)$, which is sometimes solvable for $x$ in terms of the Lambert W function.

I'm not quite sure what other problems are solvable with the Lambert W function, maybe some differential equations, but that is increasingly beyond my ability to explain or understand.