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Sibuya Y. — Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation
Sibuya Y. — Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation

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Название: Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation

Автор: Sibuya Y.


Research in differential equations is usually oriented toward explicit results and motivated by applications. Many clever methods have been discovered in this way, but, when problems of more fundamental difficulty arise, researchers must find something intrinsic in the mathematics itself in order to make progress. As research in topology, algebraic geometry, and functions of several complex variables have advanced, many methods useful in such fields were introduced into the study of differential equations.
The main part of this book is a translation of a 1976 book originally written in Japanese. The book, focusing attention on intrinsic aspects of the subject, explores some problems of linear ordinary differential equations in complex domains. Examples of the problems discussed include the Riemann problem on the Riemann sphere, a characterization of regular singularities, and a classification of meromorphic differential equations. Since the original book was published, many new ideas have developed, such as applications of D-modules, Gevrey asymptotics, cohomological methods, $k$-summability, and studies of differential equations containing parameters. Five appendices, added in the present edition, briefly cover these new ideas. In addition, more than 100 references have been added. This book will introduce readers to the essential facts concerning the structure of solutions of linear differential equations in the complex domain, as well as illuminate the intrinsic meaning of older results by means of more modern ideas. A useful reference for research mathematicians on various fundamental results, this book would also be suitable as a textbook in a graduate course or seminar.

Язык: en

Рубрика: Математика/Анализ/Дифференциальные уравнения/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1990

Количество страниц: 267

Добавлена в каталог: 16.04.2005

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Предметный указатель
$(\mathscr G, \mathscr H)$-maps      46
$(\mathscr G, \mathscr H)$-sdeformations      47
$D_{\mathscr A}$-modules      178
(mutually) nomotopic      13
Apparent singular points      110
Asymptotic expansions      117
Asymptotic expansions of Gevrey orders      190
Asymptotic systems      168
Birkhoff's lemma      75
Birkhoff's theorem      73
Cartan's Lemma      51
Cauchy integral representation      2
Characterization of regular singular points      115
Connection matrices along a curve      12
Construction of fibre bundles      64
Continuous deformations      45
Covering at $z=\infty$      144
Cross-sections of an asymptotic system      168
Cyclic vectors      126 182
Decomposition theorem of Cartan      35 47
Deformation theorem of Grauert      51
Existence theorem of Grauert      29 30 61
Existence theorem of Kimura      107
Formally equivalent      167
Formally equivalent of Gevrey order s      222
Fuchsian modules      180
Fuchsian type      102
Fundamental group      14
Fundamental solutions      26
Gevrey asymptotics      190
Holomorphic deformations      40
Holomorphic functions      1
Holomorphically convex      60
Holomorphically equivalent      167
Hukuhara — Turrittin's theorem      162
Hukuhara's theorem      89
Incomplete Leroy transforms      200
Irregularity      220
Jurkat — Lutz's theorem      140
k-summable      231
k-summable in a direction      231
Katz's result      135
Kimura's lemma      83
Lattice      180
Lin's theorem      213
Linear groups      32
Maillet's theorem      190
Majima asymptotics      240
Newton polygons      182
Normally Fuchsian type      102
Normally quasi-Fuchsian type      101
Normally regular singular points      99 102
Obstruction cocycles      65
Oka — Weil domains      60
Order of singularity of a transformation at $z=\infty$      131
Order of solutions at $z=\infty$      118
Phragmen — Lindeloef theorem in a cohomological form      214
Power series      2
Power series of Gevrey order s      190
Quasi-Fuchsian type      101
Regular singular points      73 102 118
Riemann's problem      96
Schlesinger's equations      248
Sections of a fibre bundle      65
Span of a transformation at $z=\infty$      132
Standard coordinates      33
Stein manifolds      60
Stokes phenomena      146 170
Stokes phenomena defined by differential equations      171
Stokes structures      173
Structures of analytic continuation      7
Structures of analytic continuation defined by differential equations      27
Structures of monodromy denned by differential equations      28
Structures of monodromy on $\mathscr D$      16
Structures of monodromy with base point at $z_0$      14
Sum of a k-summable power series      231
Systems of connection matrices      5 11
Systems of connection matrices denned by a system of fundamental solutions      27
Systems of fundamental solutions      26
Systems of fundamental solutions in the neighborhood of $z=\infty$      145 170
Taylor series      4
Turrittin's problem      235
Turrittin's theorem      91
Vector bundles      76
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