You find a lot of information in: 
Dmitri Alekseevky, Andreas Kriegl, Mark Losik, Peter W. Michor: Choosing roots of polynomials smoothly, Israel J. Math 105 (1998), p. 203-233. [(pdf)][1],
and in related later papers. The formula that you seek (or the inverse) is on page 7.


  [1]: http://www.mat.univie.ac.at/~michor/roots.pdf