According to this link, Lambert W function $"W_k(f(x))"$ has only 2 branches $(k=0,\,k=-1)$. But, there's an equation that has $4$ solutions ($2$ real solutions and $2$ complex solutions which is a pair of complex conjugates).
The equation is: $$\frac{x^3}{24} - \ln{x} =0$$ The solutions:
- $e^{-\frac 13 W_{-1} \left(-\frac 18\right)}$ (Real solution)
- $e^{-\frac 13 W_{0} \left(-\frac 18\right)}$ (Real solution)
- $e^{-\frac 13 W_{1} \left(-\frac 18\right)}$ (Complex solution)
- $e^{-\frac 13 W_{2} \left(-\frac 18\right)}$ (Complex solution)
And what i want to ask here is how to find those complex solutions? Yeah, algebraically i can solve that equation by myself. But, then how to determine $k=({-1,0,1,2})$? for finding the entire solutions? This question is related to the given link, but they weren't discussing about how to obtain complex solutions using this special function and how to determine the value of $k$.
I mean, consider if i have an equation that has $7$ solutions (just for a random example), and actually those solutions are obtained by determining all possibilities of the value of $k$. Again, how do we know that? Cz since we never know how to find $k$ we can't find the whole solutions.
Hope you can help me to understand. Thanks!