Correspondence Function D(z) and Generalized Irregular Equations

Extending Riemann's idea of P function (using equation's parameters to represent the function defined by the equation), we introduce correspondence functions D(z) to describe regular and irregular integrals in a unifying way. By explicit solution discuss monodromy group of non-Fuchsian equations. The explicit expressions of exponent and expansion coefficients for Floquet solution are obtained. Method of correspondence functions permits us to obtain systematically the solutions of generalized irregular equations, having regular, irregular poles, essential, algebraic, transcendental, logarithmic singularities as well as singular line. The representation of basic set of solutions by D_σ(z) function makes it possible to enlarge the scope of investigation of analytic theory. The significance of Poincare's conjecture is discussed, as D functions are automorphic.