New Theory for Equations of Non-Fuchsian Type——Representation Theorem of Tree Series Solution(Ⅱ)

Our main result consists in proving the representation theorem, Irregular integral is a new type of analytic functions.represented by a compound Taylor-Fourier tree series, in whick each coefficient of the Fourier series is a Taylor series, while the Taylor coefficients are tree series in terms of equations parameters, higher order correctibn terms to each coefficient having tree structure with inesaustalile proliferation.The solution obtained is proved to be convergent absolutely and uniformly in the region defined by coefficient functions of the original equation, provided the structure parameter is less than unity. Direct substitution shows that our tree series solution satisfies the equation ezulicitlv eeneration by eeneration.As compared with classical theory our method not only furnishes explicit expression of irregular integral, leading to the solution of Poincaré problem, but also provides possibility of extending the scope of investigation for analytic theory to equations with various kinds of singularities in a unifying way.Enact explicit analytic expression for irregular integrals can be obtained by means of correspondence principle.It is not difficult to prove the convergence of the tree series solution obtained. Direct subsitution shows it satisfies the equation.The tree series is automorphic, which agrees completely with Poincaré's conjecture.