Solving $W$ function What is the general form of Lambert $W$-function to calculate any $x$ in $\mathbb R$?
I had problems solving for $x, x^2-e^x=0$.
I reached $x=-2W(\frac12)$.
What does $W(\frac12)$ equal to?

There is no "general form of the Lambert W-function". All you have is the definition, tables, power series, other numerical methods. It's like the trigonometric functions - no algebraic "formula". 
From wikipedia:

The Lambert W relation cannot be expressed in terms of elementary
  functions.1

Nowadays the trig functions are so common that you don't tend to think of them as special in that way. 
The Lambert W-function seems to appear more and more often on this site. Perhaps some day people will view it as just another special function.
