FORMAL POWER SERIES AND LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS

By F. FAVAM

CONTENTS BASIC PROPERTIES OF SOLUTIONS SIMPLY CONNECTED REGIONS FUNDAMENTAL SOLUTIONS SYSTEMS IN GENERAL REGIONS INHOMOGENEOUS SYSTEMS REDUCED SYSTEMS SOME ADDITIONAL NOTATION SINGULARITIES OF FIRST KIND SYSTEMS WITH GOOD SPECTRUM CON?UENT HYPERGEOMETRIC SYSTEMS HYPERGEOMETRIC SYSTEMS SYSTEMS WITH GENERAL SPECTRUM SCALAR HIGHER-ORDER EQUATIONS HIGHEST-LEVEL FORMAL SOLUTIONS FORMAL TRANSFORMATIONS THE SPLITTING LEMMA NILPOTENT LEADING TERM TRANSFORMATION TO RATIONAL FORM ASYMPTOTIC POWER SERIES SECTORS AND SECTORIAL REGIONS FUNCTIONS IN SECTORIAL REGIONS FORMAL POWER SERIES ASYMPTOTIC EXPANSIONS GEVREY ASYMPTOTICS GEVREY ASYMPTOTICS IN NARROW REGIONS GEVREY ASYMPTOTICS IN WIDE REGIONS INTEGRAL OPERATORS LAPLACE OPERATORS BOREL OPERATORS INVERSION FORMULAS A DI?ERENT REPRESENTATION FOR BOREL OPERATORS GENERAL INTEGRAL OPERATORS KERNELS OF SMALL ORDER PROPERTIES OF THE INTEGRAL OPERATORS CONVOLUTION OF KERNELS SUMMABLE POWER SERIES GEVREY ASYMPTOTICS AND LAPLACE TRANSFORM SUMMABILITY IN A DIRECTION ALGEBRA PROPERTIES DE?NITION OF K-SUMMABILITY GENERAL MOMENT SUMMABILITY FACTORIAL SERIES CAUCHY-HEINE TRANSFORM DE?NITION AND BASIC PROPERTIES NORMAL COVERINGS DECOMPOSITION THEOREMS FUNCTIONS WITH A GEVREY ASYMPTOTIC SOLUTIONS OF HIGHEST LEVEL THE IMPROVED SPLITTING LEMMA MORE ON TRANSFORMATION TO RATIONAL FORM SUMMABILITY OF HIGHEST-LEVEL FORMAL SOLUTIONS FACTORIZATION OF FORMAL FUNDAMENTAL SOLUTIONS DE?NITION OF HIGHEST-LEVEL NORMAL SOLUTIONS STOKES’ PHENOMENON HIGHEST-LEVEL STOKES’ MULTIPLIERS THE PERIODICITY RELATION THE ASSOCIATED FUNCTIONS AN INVERSION FORMULA COMPUTATION OF THE STOKES MULTIPLIERS HIGHEST-LEVEL INVARIANTS THE FREEDOM OF THE HIGHEST-LEVEL INVARIANTS 10 MULTISUMMABLE POWER SERIES 10 1 CONVOLUTION VERSUS ITERATION OF OPERATORS 10 2 MULTISUMMABILITY IN DIRECTIONS 10 3 ELEMENTARY PROPERTIES 10 4 THE MAIN DECOMPOSITION RESULT 10 5 SOME RULES FOR MULTISUMMABLE POWER SERIES 10 6 SINGULAR MULTIDIRECTIONS 10 7 APPLICATIONS OF CAUCHY-HEINE TRANSFORMS 10 8 OPTIMAL SUMMABILITY TYPES 11 ECALLE’S ACCELERATION OPERATORS 11 1 DE?NITION OF THE ACCELERATION OPERATORS 11 2 ECALLE’S DE?NITION OF MULTISUMMABILITY 11 3 CONVOLUTIONS 11 4 CONVOLUTION EQUATIONS 12 OTHER RELATED QUESTIONS 12 1 MATRIX METHODS AND MULTISUMMABILITY 12 2 THE METHOD OF REDUCTION OF RANK 12 3 THE RIEMANN-HILBERT PROBLEM 12 4 BIRKHO?’S REDUCTION PROBLEM 12 5 CENTRAL CONNECTION PROBLEMS 13 APPLICATIONS IN OTHER AREAS, AND COMPUTER ALGEBRA 13 1 NONLINEAR SYSTEMS OF ODE 13 2 DI?ERENCE EQUATIONS 13 3 SINGULAR PERTURBATIONS 13 4 PARTIAL DI?ERENTIAL EQUATIONS 13 5 COMPUTER ALGEBRA METHODS 14 SOME HISTORICAL REMARKS SOLUTIONS TO THE EXERCISES REFERENCES