Say I have an equation that I can solve in $x$ as follows:
$$ x = LambertW_{-1}(y)$$
Where LambertW is the product-log function.
Can I say I have an explicit solution for $x$? It looks like that, but I'm not sure, given that I used the product-log function to solve for $ye^y$ - something which doesn't have an explicit solution in my books.
Is using the product-log function just cheating? Is it just a list of numerical values saved somewhere? I know that the derivatives are well-defined in term of other functions, but that's not really sufficient.
Define explicit solution
How about we call it closed-form solution then. To quote Wolfram Alpha:
An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally-accepted set. For example, an infinite sum would generally not be considered closed-form. However, the choice of what to call closed-form and what not is rather arbitrary since a new "closed-form" function could simply be defined in terms of the infinite sum.
Does Lambert-W belong into this "generally-accepted" set?
First of all, an explicit solution is not so well defined from you, so I will try to do my best to satisfy your question.
Maybe, but if it is cheating, aren't logarithms cheating? Isn't the entire base of trigonometry just functions we made up for convenience? In essence, I could 'invent' a function $f(x)$ that was the inverse of $g(x)=x+\sin(x)$, which naturally doesn't exist using a finite amount of known functions. If people found it useful enough, then maybe this function will get a name and people will use it.
The Lambert W function was just as similar. Someone (Lambert W) had a problem involving exponential functions and he couldn't solve it using known functions. So, Lambert W simply 'made' a new function, the Lambert W/product log function. Later, people saw that it was useful and had applications, so now this function is more 'official' and more accepted.
No, you can actually approximate the product log using linear approximation, for example. I would think saving some numerical values, including complex values, would be too tedious. You should be able to find many online calculators that can solve for complex inputs and even a choice of which branch of the product log you want.