Some Historical Remarks

Abstract

Since the middle of the last century, mathematicians and physicists alike have observed that simple ODE may have solutions in terms of power series whose coefficients grow at such a rate that the series has a radius of convergence equal to zero. In fact, it became clear later that every linear meromorphic system, in the vicinity of a so-called irregular singularity, has a formal fundamental solution of a certain form. This “solution” can be relatively easily computed, in particular nowadays, where one can use computer algebra packages, but it will, in general, involve power series diverging everywhere. It was quite an achievement when it became clear that formal, i.e., everywhere divergent, power series solving even nonlinear meromorphic systems of ODE can be interpreted as asymptotic expansions of certain solutions of the same system. This by now classical theory has been presented in many books on differential equations in the complex plane or related topics. The presentations I am most familiar with are the monographs of Wasow [281] and Sibuya [251]. The most important result in this context is, in Wasow’s terminology, the Main Asymptotic Existence Theorem: It states that to every formal solution of a system of meromorphic differential equations, and every sector in the complex plane of sufficiently small opening, one can find a solution of the system having the formal one as its asymptotic expansion. This solution, in general, is not uniquely determined, and the proofs given for this theorem, in various degrees of generality, do not provide a truly efficient way to compute such a solution, say, in terms of the formal solution.