I’m very interested in Lambert W function and I want to know how to check if some equation can be solved using this function.
For this equation it is obviously that x=Wk(a) where k∈Z.
Now we must reduce it to form ef(x)f(x)=c and then use Lambert W function.
This is not too hard to solve.
It is very hard to solve this and after long computation we will get
There is no known solution for this equation.
My question is how to check if some equation can be solved using Lambert W function.
Assume an ordinary equation F(x)=c is given where c is a constant and F is a function. Isolating x on one side of the equation only by operations to the whole equation means to apply a suitable partial inverse function (branch of the inverse relation) F−1 of F: x=F−1(c).
The problem of existence of elementary inverses of elementary functions is solved by the theorem in [Ritt 1925] that is proved also in [Risch 1979].
The problem of existence of elementary numbers as solutions of irreducible polynomial equations P(x,ex)=0 is solved in [Lin 1983] and [Chow 1999].
But LambertW is not an elementary function.
I. a. the following kinds of equations of x can be solved in closed form by applying Lambert W or without Lambert W.
f,f1 functions in C with suitable local closed-form inverses.
If c2,c4≠0, x=0 and f(x)=0 respectively is a solution.
If you have an x-containing summand on the left-hand side of the equation, you can subtract the exponential term from the equation and divide the equation by it to get a product of a power of x (or f(x)) and an exponential term of x (or f(x)) to apply Lambert W.
See also [Edwards 2020], [Galidakis/ Weisstein], [Köhler].
If your equation contains an exponential function or a logarithm function but cannot be brought to the form of equations (1) – (4), you could try to apply a predefined generalization of Lambert W.
See e.g. [Corcino/Corcino/Mezö 2017], [Dubinov/Galidakis 2007], [Galidakis 2005], [Maignan/Scott 2016], [Mezö 2017], [Barsan 2018].
[Corcino/Corcino/Mezö 2017] Corcino, C. B.; Corcino, R. B.; Mezö, I.: Integrals and derivatives connected to the r-Lambert function. Integral Transforms and Special Functions 28 (2017) (11)