Singularities of First Kind

Abstract

Throughout this chapter, we shall be concerned with a system (1.1) (p. 2) having a singularity of first kind, i.e., a pole of first order, at some point z ₀ , and we wish to study the behavior of solutions near this point. In particular, we wish to solve the following problems as explicitly as we possibly can:

• P1) Given a fundamental solution X(z) of (1.1), find a monodromy matrix at z ₀ ; i.e., find M so that X(z) = S(z) (z - z _o )^M, with S(z) holomorphic and single-valued in 0 <

• P2) Determine the kind of singularity that S(z) has at z _o ; i.e., decide whether this singularity is removable, or a pole, or an essential one.

• P3) Find the coefficients in the Laurent expansion of S(z) about the point z ₀ , or more precisely, find equations that allow the computation of at least finitely many such coefficients.