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A Direct Proof that Solutions of the Six Painlevé Equations Have No Movable Singularities Except Poles
Author(s) -
Joshi Nalini,
Kruskal Martin D.
Publication year - 1994
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1994933187
Subject(s) - gravitational singularity , mathematics , logarithm , property (philosophy) , algebraic number , mathematical analysis , differential equation , pure mathematics , direct proof , order (exchange) , philosophy , epistemology , finance , economics
The Painlevé property of an n th‐order differential equation is that no solution has any movable singularities other than poles. This property is strongly indicative of complete integrability (the existence of n − 1 integrals). However, the usual technique employed to test for the Painlevé property seeks only movable algebraic (or logarithmic) singularities. More general singularities are ignored. But, the six standard Painlevé equations are known to have no such singularities. Painlevé's proof of this is long and laborious; we give here a direct proof.