Basic Properties of Solutions.- Singularities of First Kind.- Highest-Level Formal Solutions.- Asymptotic Power Series.- Integral Operators.- Summable Power Series.- Cauchy-Heine Transform.- Solutions of Highest Level.- Stokes' Phenomenon.- Multisummable Power Series.- Ecalle's Acceleration Operators.- Other Related Questions.- Applications in Other Areas, and Computer Algebra.- Some Historical Remarks.

Simple Ordinary Differential Equations 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, every linear meromorphic system has a formal solution of a certain form, which can be relatively easily computed, but which generally involves such power series diverging everywhere. In this book the author presents the classical theory of meromorphic systems of ODE in the new light shed upon it by the recent achievements in the theory of summability of formal power series.