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Singularity Analysis and a Function Unifying the Painlevé and the Psi Series

Abstract The classical (ARS) algorithm used in the Painlevé test picks up only those functions analytic in the complex plane. We complement it with an iterative algorithm giving the leading order and the next terms in all cases. This algorithm works both for an ascending series (about a singularity at finite time) and a descending series (asymptotic expansion for t → ∞). The algorithm introduces naturally the logarithmic terms when they are necessary. The calculation, given in the first place for a system possessing the two symmetries of time translation and self-similarity, is subsequently generalised to the case in which this last symmetry is broken. The algorithm enlarges the class of equations for which more explicit methods (Lie symmetries, Darboux and Carleman invariants etc) should be applied with a certain hope of success.