New theory for equations of non-fuchsian type —Representation theorem of tree series solution (II)

Abstract

Our main result consists in proving the representation theorem. Irregular integral is a new type of analytic function, represented by a compound Taylor-Fourier tree series, in which each coefficient of the Fourier series is a Taylor series, while the Taylor coefficients are tree series in terms of equations parameters, higher order correction terms to each coefficient having tree structure with inexhaustible 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 explicity generation by generation.

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.

Exact 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 substitution shows it satisfies the equation.

The tree series is automorphic, which agrees completely with Poincaré's conjecture.