01270nas a2200109 4500008004100000245012700041210007000168260002800238520083700266100002101103856003601124 2012 en d00aSolving the Sixth PainlevĂ© Equation: Towards the Classification of all the Critical Behaviors and the Connection Formulae0 aSolving the Sixth PainlevĂ© Equation Towards the Classification o bOxford University Press3 aThe critical behavior of a three real parameter class of solutions of the\\r\\nsixth Painlev\\\\\\\'e equation is computed, and parametrized in terms of monodromy\\r\\ndata of the associated $2\\\\times 2$ matrix linear Fuchsian system of ODE. The\\r\\nclass may contain solutions with poles accumulating at the critical point. The\\r\\nstudy of this class closes a gap in the description of the transcendents in one\\r\\nto one correspondence with the monodromy data. These transcendents are reviewed in the paper. Some formulas that relate the monodromy data to the critical behaviors of the four real (two complex) parameter class of solutions are\\r\\nmissing in the literature, so they are computed here. A computational procedure to write the full expansion of the four and three real parameter class of solutions is proposed.1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/6093