New Development in Poincaré’s Problem of Irregular Integrals

In connection with non-Fuchsian equations Poincaré has made an important conclusion; 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 of which can be represented by conventional recursive series, while its remaining part is expressed 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 classical theory (Hill-Poincaré-von Koch),our method yields naturally exact analytic solution in explicit form, The method proposed map be used to;construct a unifying theory for general equations with variable coefficients, having varioas kinds of singularities as singular lines. The significance of Poincaré conjecture is discussed. The tree series obtained belong to higher automorphic functions.