Global Analysis in Linear Differential Equations - M. Kohno

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Résumé :
This book treats several topics on the global analysis of linear differential equations in the complex domain, using original methods of analysis developed by K. Okubo and the present author. For example, global analysis of linear differential equations is closely related to that of linear difference equations. Through the asymptotic behaviour of solutions of linear difference equations, connection problems for hypergeometric systems can be solved, i.e. Fuchsian differential equations and Stokes phenomenon for Birkhoff systems. Furthermore, the calculation of monodromy groups is explained in detail. br/ emAudience:/em This volume will be of interest to researchers and graduate students whose work involves ordinary differential equations, special functions, finite difference, functional equations, approximations and expansions, and symbolic/algebraic manipulation.

Sommaire:
1 Preparations.- 1.1 Convergent and Divergent Series.- 1.2 Asymptotic Expansions.- 1.3 Linear Difference Equations.- 1.4 Hypergeometric Difference Equation.- 1.5 Modified Gamma Function.- 2 Gauss and Airy Equations.- 2.1 Gauss Equation.- 2.2 Rummer's Connection Formulas.- 2.3 Monodromy Groups.- 2.4 Associated Fundamental Function.- 2.5 Airy Equation.- 3 Linear Differential Equations.- 3.1 Remarks on Holomorphic Functions.- 3.2 Existence Theorems of Differential Equations.- 3.3 Classification of Singularities.- 3.4 Regular Singular Point.- 3.5 Irregular Singular Point.- 4 Reduction Problems.- 4.1 Reduction to Hypergeometric System.- 4.2 Reduction to Birkhoff Canonical System.- 4.3 Algebraic Manipulation.- 5 Monodromy Groups for Hypergeometric Systems.- 5.1 Extended Gauss Formula.- 5.2 Calculation of Monodromy Groups.- 5.3 Monodromy Group in Logarithmic Case.- 6 Connection Problem for Hypergeometric Systems.- 6.1 General Theory.- 6.2 H. Galbrun Theory.- 6.3 Hierarchy of Connection Coefficients.- 6.4 Jordan-Pochhammer Equation.- 6.5 Appell Hypergeometric Functions.- 6.6 Frobenius Theorem.- 7 Stokes Phenomenon.- 7.1 Two Point Connection Problem.- 7.2 Associated Fundamental Function.- 7.3 Extended Bessel Equation.- 7.4 Derivatives of Stokes Multipliers.- 7.5 Multi-point Connection Problem.