Preface The Gamma function, the zeta function, the theta function, the hyper geometric function, the Bessel function, the Hermite function and the Airy function, . . . are instances of what one calls special functions. These have been studied in great detail. Each of them is brought to light at the right epoch according to both mathematicians and physicists. Note that except for the first three, each of these functions is a solution of a linear ordinary differential equation with rational coefficients which has the same name as the functions. For example, the Bessel equation is the simplest non-trivial linear ordinary differential equation with an irreg ular singularity which leads to the theory of asymptotic expansion, and the Bessel function is used to describe the motion of planets (Kepler's equation). Many specialists believe that during the 21st century the Painleve functions will become new members of the community of special func tions. For any case, mathematics and physics nowadays already need these functions. The corresponding differential equations are non-linear ordinary differential equations found by P. Painleve in 1900 fqr purely mathematical reasons. It was only 70 years later that they were used in physics in order to describe the correlation function of the two dimen sional Ising model. During the last 15 years, more and more people have become interested in these equations, and nice algebraic, geometric and analytic properties were found.

Autorentext
Masaaki Yoshida ist Professor für Mathematik an der Kyushu Universität, Japan.

Klappentext

This book gives an introduction to the modern theory of special functions. It focuses on the nonlinear Painlevé differential equation and its solutions, the so-called Painlevé functions. It contains modern treatments of the Gauss hypergeometric differential equation, monodromy of second order Fuchsian equations and nonlinear differential equations near singular points.The book starts from an elementary level requiring only basic notions of differential equations, function theory and group theory. Graduate students should be able to work with the text."The authors do an excellent job of presenting both the historical and mathematical details of the subject in a form accessible to any mathematician or physicist." (MPR in "The American Mathematical Monthly" März 1992.

Inhalt

Elements of Differential Equations.- 1.1 Cauchy's existence theorem.- 1.2 Linear equations.- 1.3 Local behavior around regular singularities (Frobenius's method).- 1.4 Fuchsian equations.- 1.5 Pfaffian systems and integrability conditions.- 1.6 Hamiltonian systems.- 2. The Hypergeometric Differential Equation.- 2.1 Definition and basic facts.- 2.2 Contiguity relations.- 2.3 Integral representations.- 2.4 Monodromy of the hypergeometrie equation.- 3. Monodromy-Preserving Deformation, Painlevé Equations and Garnier Systems.- 3.1 Painlevé equations.- 3.2 The Riemann-Hilbert problem for second order linear differential equations.- 3.3 Monodromy-preserving deformations.- 3.4 The Garnier system 𝒢n.- 3.5 Schlesinger systems.- 3.6 The Schlesinger system and the Garnier system 𝒢n.- 3.7 The polynomial Hamiltonian system ?nassociated with 𝒢n.- 3.8 Symmetries of the Garnier system 𝒢nand of the system ?n.- 3.9 Particular solutions of the Hamiltonian system ?n.- 4. Studies on Singularities of Non-linear Differential Equations.- 4.1 Singularities of regular type.- 4.2 Fixed singular points of regular type of Painlevé equations.- Notes on the chapter titlepage illustrations.- Index of symbols.