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Balser W. — Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations
Balser W. — Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations

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Название: Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations

Автор: Balser W.

Аннотация:

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.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2000

Количество страниц: 299

Добавлена в каталог: 14.12.2009

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Admissible      165
Admissible, multidirection      161
Algebra      215
Algebra with unit element      215
Algebra, commutative      215
Algebra, differential      216
Anti-Stokes' direction      137
Associated function      146
Asymptotic equality      65
Asymptotic expansion      65
Asymptotic expansion, remainders of      65
Asymptotic of Gevrey order      70
Asymptotic space      75
Banach space      216
Bessel's equation      24
Bessel's function      23
Beta integral      228
Block structure of highest level      134
Block structure of level $\mu$      134
Blocked, diagonally      214
Blocked, triangularly      214
Blocks, direct sum of      214
Blocks, Jordan      211
Borel operator      80
Borel operator, formal      80
Closed sector      60
Closed w.r. to continuation      57 132
Complex plane, compactified      15
Confluent hypergeometric function      23
Confluent hypergeometric function, generalized      24
Confluent hypergeometric, equation      24 25
Connection matrix      193
Connection problem      193
Convergence compact      243
Convergence compact, locally uniform, of an integral      223
Convolution      178
Convolution of kernel functions      160
Convolution, formal      178
Curve      62
Data pairs of FFS      132
Data pairs of FFS, ordered      134
Derivation      216
Differentiability at the origin      62
Differential algebra      216
Differential field      73
Direction, anti-Stokes'      137
Direction, Stokes'      137
Distinct eigenvalue case      44
Entire function      233
Equation, Bessel's      24
Equation, hypergeometric      26
Equation, hypergeometric, confluent      24 25
Equation, hypergeometric, generalized      27
Equation, indicial      36
Equation, Legendre's      5
Equivalence, analytic      28 153
Equivalence, meromorphic      153
Exponential growth      62
Exponential growth in a direction      63
Exponential order      232
Exponential shift      39
Factorial series      110
Finite type      233
Floquet exponent      20
Floquet solution      20
Floquet solution for scalar ODE      34
Formal fundamental solution      131
Formal fundamental solution of highest level      55
Formal Laplace operator      79
Formal Laurent series      72 124
Formal power series      64
Formal solution      42
Formal system      40
Formal system of Gevrey order s      40
Formula, Hankel's      228
Formula, Stirling's      229
Fuchsian system      188
Function of a matrix      237
Function, associated      146
Function, Bessel's      23
Function, bounded at the origin      61
Function, continuous at the origin      61
Function, entire      233
Function, Gamma      227
Function, hypergeometric      26
Function, hypergeometric, confluent      23
Function, hypergeometric, generalized      26
Function, kernel      85 89
Function, Kummer's      23
Function, matrix-valued      2
Function, Mittag — Leffler's      233
Function, Mittag — Leffler's, aymptotic of      84
Function, order of a      232
Function, root of a      230
Function, single-valued      227
Function, type of a      233
Fundamental solution      5
Fundamental solution, computation of      6
Fundamental solution, existence of      6
Fundamental solution, formal      42
General summability method      97
Gevrey order of a system      40
Gevrey order of a transformation      39
Gevrey order, asymptotic      70
Good spectrum      18
Hankel's formula      228
Highest level, normal solution      138
HLFFS, data pairs of a      56
Holomorphic continuation      224
Holomorphic matrix-valued f.      2
Holomorphy in Banach spaces      219
Holomorphy, weak      219
Homotopic paths      225
Hypergeometric equation      26
Hypergeometric equation, confluent      24
Hypergeometric equation, generalized      27
Hypergeometric function      26
Hypergeometric function, confluent      23
Hypergeometric function, generalized      26
Hypergeometric series, generalized      107
Hypergeometric system      25
Hypergeometric system, confluent      21
Indicial equation      36
Integrability at the origin      86
Invariants      153
Jordan block      211
Jordan Matrix      211
k-summability      105
k-summability at infinity      123
k-summability in a direction      100
Kernel function      85
Kernel functions, convolution of      160
Kernel of small order      89
Laplace operator      78
Laplace operator, finite      74
Laplace operator, formal      79
Laurent series, formal      124
Leading term      37
Legendre equation      5
Legendre polynomials      5
Level, highest      134
Level, set      134
Logarithm, Riemann surface of the      226
Matrix, canonical form of a      211
Matrix, companion      4 52
Matrix, connection      193
Matrix, Jordan      211
Matrix, method      97
Matrix, monodromy      9
Matrix, nilpotent, order of a      212
Matrix, permutation      143
Matrix, power series      3
Matrix, spectrum of a      211
Matrix, unit      211
Moderate growth      55
Moebius transformation      15
Moment methods      107
Moment sequence      119
Moment, condition      86
Moment, function      86
Moments      107
Monodromy factor      9
Monodromy factor, formal      143
Monodromy matrix      9
Monodromy matrix, formal      143
Multi-index      198
Multidirection      161
Multidirection, singular      167
Multidirection, singular, of level j      168
Multisummability      173
Multisummability, type of      165
Nilpotent matrices, superiority      45
Norm      216 292
Normal covering      118
Operator, acceleration      176
Operator, acceleration, formal      176
Operator, Borel      80
Operator, Borel, formal      80
Operator, deceleration      177
Operator, Laplace      78
Operator, Laplace, finite      74
Operator, Laplace, formal      79
Order of a root      230
Order of kernel functions      86
Order of moments      86
Order, exponential      232
Paths      62
Paths, homotopic      225
Pochhammer symbol      22
Poincare rank      14
Power series regularity      183
Power series, matrix      3
Power series, vector      3
Principal value of log Z      9
Problem, connection      193
Problem, connection, central      193
Problem, connection, lateral      193
Projection      226
Punctured disc      8
Punctured plane      226
Rank, Poincare      14
Reduced system      12
Reduced system, type of      12
Region      219
Region, sectorial      60
Region, sectorial, at infinity      123
Region, simply connected      1
Regular-singular point      33
Regular-singular point for scalar equations      34
Regular-singular point, almost      55
Residue      224
Riemann surface of the logarithm      226
Ring      8
Root of a function      230
Root, order of a      230
Sector      60
Sector, bisecting direction of a      60
Sector, closed      60
Sector, opening of a      60
Sector, radius of a      60
Sectorial region      60
Sectorial region at infinity      123
Series, convergent      64
Series, formal      64
Series, formal Laurent      72
Series, Laurent      224
Series, sum of a      64
Similarity      211
Singular set      168
Singular, multidirection      167
Singularity of first kind      14
Singularity of second kind      14
Singularity of the first kind for scalar equations      34
Singularity, apparent      16
Singularity, essential      224
Singularity, essentially irregular      55
Singularity, pole      224
Singularity, regular      33
Singularity, regular, almost      55
Singularity, regular, for scalar equations      34
Singularity, removable      224
Solution, Floquet      20
Solution, Floquet, for scalar ODE      34
Solution, fundamental      5
Solution, fundamental, computation of      6
Solution, fundamental, existence of      6
Solution, fundamental, formal      42 131
Solution, fundamental, of highest level      55
Solution, logarithmic      35
Solution, normal, of highest level      138
Space, asymptotic      75
Spectrum      211
Spectrum, good      18
Stirling's formula      229
Stokes' direction      137
Stokes' multipliers of highest level      140
Sum of a series      64
Summability factor      109
Summability of series in roots      124
Summability, domain      97
Summability, method, general      97
Summability, method, matrix      97
System, elementary      14
System, formal      40
System, formal, of Gevrey order s      40
System, Fuchsian      188
System, hypergeometric      25
System, hypergeometric, confluent      21
System, meromorphic      14
System, nonlinear      198
System, nonlinear, homogeneous      198
System, normalized      145
System, normalized, w. nilpotent l.t.      47
System, rank-reduced      187
System, reduced      12
System, reduced, type of      12
System, reduced, w. nilpotent l.t.      47
System, transformed      28 39
Transform, Cauchy — Heine      116
Transform, Cauchy — Heine, formal      117
Transformation, analytic      27
Transformation, analytic, at infinity      38
Transformation, analytic, of Gevrey order s      39
Transformation, constant      27
Transformation, exponential shift      39
Transformation, formal analytic, at infinity      38
Transformation, meromorphic      39
Transformation, meromorphic, formal      39
Transformation, Moebius      15
Transformation, q-analytic      39
Transformation, q-meromorphic      39
Transformation, shearing      39
Transformation, shearing, unramified      39
Transformation, terminating      38
Transformed system      28 39
TYPE      233
Type, finite      233
Unit element      215
Variation of constants      11
Vector, cyclic      5
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