We consider single differential equations on the complex projective line which have one regular singular point and one irregular singular point. For any sector, any fundamental set of solutions near the regular singular point, and any fundamental set of solutions near the irregular singular point on the sector, there is a linear transformation relating the two fundamental sets of solutions. The problem of finding the coefficients of this linear transformation is called the connection problem. By composing these linear transformations, we can analyze the Stokes phenomenon. We construct a family of functions whose asymptotic expansions match those of a fundamental solution at a regular singular point. These functions are particular solutions of first order nonhomogeneous differential equations that can be derived from the fundamental solutions at the regular singular point and formal solutions at the irregular singular point of the original differential equation. We call these functions the fundamental functions associated with this two point connection problem. The series expansions of the associated fundamental functions are described by systems of difference equations, and the coefficients relating them to the fundamental solutions can be found by a recursive process. This yields a method for calculating the linear relation between the two fundamental sets of solutions. In this talk, we introduce the Okubo-Kohno method to solve the two point connection problem for linear ordinary differential equations with an irregular singularity of rank one by analyzing the associated fundamental functions. We will also apply this method to the connection problem with more than two singular points.
Seminar on Analysis at University of Tsukuba