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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.
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.
Multisummability is a method which, for certain formal power series
with radius of convergence equal to zero, produces an analytic
function having the formal series as its asymptotic expansion. This
book presents the theory of multisummabi- lity, and as an
application, contains a proof of the fact that all formal power
series solutions of non-linear meromorphic ODE are multisummable.
It will be of use to graduate students and researchers in
mathematics and theoretical physics, and especially to those who
encounter formal power series to (physical) equations with rapidly,
but regularly, growing coefficients.
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