I mean functions defined by power series (just like sin(x) is defined in analysis courses). For the three body problem, see http://oro.open.ac.uk/22440/2/Sundman_final.pdf (Warning, pdf!). This is what Wikipedia cites when talking about the solution to the three body problem. The document gives a lout of historical context.

For polynomial roots, see wikipedia.org/wiki/Elliptic_function.