NEW DEVELOPMENT IN POINCAR’S PROBLEM OF IRREGULAR INTEGRALS

In connection with non-Fuchsian equations Poincaré has made an importantconclusion; It is impossible to obtain explicit expressions of irregular integrals(?).To elucidate the essence of Poincaré’s problem. we establish correspondence theorem.Irregular integrals are analytic functions of new kind, possessing tree structure, part ofwhich can be represented by conventional recursive series.while its remaining part isexpressed by the so-called tree series, not subjecting to any recursive relation at all.In contrast to the numerical solution calculated by infinite determinant of classicaltheory (Hill-Poincaré-von Koch), our method yields naturally exact, analytic solution inexplicit form. The method proposed may be used to construct a unifying theory for generalequations with variable coefficients. having various kinds of singularities as singular lines.The significance of Poincaré conjecture is discussed, the tree series obtained belong tohigher automorphic functions.