Lambert W function, W(x), representation for entire domain

Question

The Taylor series for the Lambert W function is $W_0(x)=\sum_{n=1}^\infty\frac{(-n)^{n-1}x^n}{n!},\left\{x\in\mathbb{R}:|x|<\frac{1}{e}\right\}$. Is there any exact closed form way to express $W(x)$ for $\left\{x\in\mathbb{R}:x\geq-\frac{1}{e}\right\}$? I am aware of Newton's method's approximation, $W(x)\approx\lim_{n\to\infty}w_n,w_{n+1}=\frac{xe^{-w_n}+(w_n)^2}{w_n+1}$, but this is not an exact value or a closed form, like the Taylor series. Also, I came across these two questions, Are there other power series for the Lambert W function than this one? and Question about Lambert W function, with comments and answers that seem to answer my question, but I am lost on how they derived them, if they are actually valid, and what the solution's closed forms are. I apologize if I sound unreasonably ignorant in this question. Thanks in advance.