Is there a general method to get the solution of
$$ xf(x)=a,$$
where $f(x)$ is an smooth analytic function?
So the solution can be expressed as $ g(a)$ and $g(a)f(g(a))=a $ for some function $g(x)$ which has a power expansion $ g(x)= \sum_{n\ge0}c_{n}x^n$ at the origin.
For example if $ f(x)=\exp(x) $ we have the Lambert function.
My answer is for solutions in closed form.
$$xf(x)=a\tag{1}$$ $$F(x)=xf(x)$$
If $f$ is an algebraic function, equation (1) is an algebraic equation and we can use the corresponding solution methods.
If $f$ is a transcendental function, we don't know how to solve the equation for $x$ by rearranging by applying only finite numbers of elementary functions we can read from the equation.
If there don't exist a representation for the function $F$ as a composition of elementary functions, $F$ doesn't have partial inverses that are elementary functions, and we need non-elementary functions therefore.
Lambert W and its similar functions are the oldest and simplest of the non-elementary functions for your problem.
Today we also have Generalized Lambert W and Hyper Lambert W.
and the functions of [Hector Vazquez-Leal / Mario Alberto Sandoval-Hernandez / Uriel Filobello-Ninoa 2020]:
What are the closed-form inverses of $x+\sinh(x)$, $x+\cosh(x)$?
