My Springer books

Table of Contents:

  • 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 Remark


  • Matrices and Vector Spaces
  • Functions with Values in Banach Spaces
  • Functions of a Matrix

From the back cover: Simple ordinary differential equations may have solutions in terms of power series whose coefficients grow at such a rate that the series have a radius of convergence equal to zero. In fact, every linear meromorphic system has a formal fundamental solution of a certain form that can be relatively easily computed, but which generally involves such power series diverging everywhere. In this book, I present the classical theory of meromorphic ODEs in the new light shed upon it by the recent achievements in the theory of summability of formal power series.

To order from the publisher, click on the scan of the book cover that you see above!

This is my earlier book of 1994, presenting the theory of multisummability founded by J. Ecalle. In particular, it contains a new proof for the main result in this area, saying that every formal solution of a non-linear meromorphic ODE is multisummable. The first proof for this was given by B. Braaksma in 1992, followed by a second one by Ramis and Sibuya in 1994. So by now three proofs for this result exist, which according to some colleagues is necessary for a theorem to be accepted as correct. All three proofs are essentially different, and my proof was judged as "tres elegante" in a survey article of B. Malgrange entitled "Sommation des series divergentes", in Expositiones Mathematicae of 1995.

This text is again available, both in electronic as well as printed form. To find out details, click on the scan of the book cover, or consult the publisher directly!