From Gauss to Painlevé

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

1. 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.- 2.1.1 The Gauss hypergeometric equation.- 2.1.2 Hypergeometric series.- 2.1.3 Finite group action and Rummer's 24 solutions.- 2.2 Contiguity relations.- 2.2.1 Contiguity relations.- 2.2.2 Contiguity relations and particular solutions of the Toda equation.- 2.3 Integral representations.- 2.3.1 Integral representations as a tool for global problems.- 2.3.2 Euler integral representation derived from the power series.- 2.3.3 The Euler transform.- 2.3.4 The hypergeometric Euler transform.- 2.3.5 Barnes integral representation: interpolation method.- 2.3.6 Barnes integral representation: difference equation method.- 2.3.7 The Gauss-Kummer identity.- 2.4 Monodromy of the hypergeometrie equation.- 2.4.1 Fundamental group of ?1 \ {0, 1, ?} and the monodromy of the Riemann equation.- 2.4.2 Classification of 2-dimensional representations of the free group with 2 generators.- 2.4.3 Finding the monodromy by local properties and the Fuchs relation.- 2.4.4 Finding the monodromy by Euler integrals over arcs.- 2.4.5 Finding the monodromy by Euler integrals over double loops.- 2.4.6 Finding the monodromy by Barnes integrals.- 2.4.7 Finding the monodromy by Gauss-Kummer's identity.- 3. Monodromy-Preserving Deformation, Painlevé Equations and Garnier Systems.- 3.1 Painlevé equations.- 3.1.1 Historical remarks.- 3.1.2 Relations between the PJ's.- 3.1.3 Symmetry of the Painlevé equation PVI.- 3.1.4 Solutions of PVI at singular points.- 3.1.5 Hamiltonian structure for PVI.- 3.1.6 Particular solutions of the PJ's.- 3.2 The Riemann-Hilbert problem for second order linear differential equations.- 3.2.1 Spaces of Fuchsian differential equations and those of representations of ?1.- 3.2.2 The Riemann-Hilbert problem.- 3.3 Monodromy-preserving deformations.- 3.3.1 M-invariant fundamental solutions.- 3.3.2 Totality of M-invariant fundamental solutions.- 3.3.3 Monodromy-preserving deformation of second order differential equations.- 3.3.4 SL-equations.- 3.3.5 Deformation equations for second order SL-equations.- 3.4 The Garnier system 𝒢n.- 3.4.1 Main theorem.- 3.4.2 Reduction to SL-equations.- 3.4.3 Explicit forms of Ki and Li.- 3.4.4 Explicit expression of Ai(x, t).- 3.4.5 Proof of Theorem 4.2.2.- 3.5 Schlesinger systems.- 3.6 The Schlesinger system and the Garnier system 𝒢n.- 3.6.1 Transformation of systems of equations into second order equations.- 3.6.2 Transformation of the Schlesinger system into the Garnier system.- 3.6.3 Transformation of second order equations into systems of equations.- 3.6.4 Relation between the Garnier system and the Schlesinger system.- 3.7 The polynomial Hamiltonian system ?nassociated with 𝒢n.- 3.7.1 Transformation of 𝒢ninto the polynomial Hamiltonian system ?n.- 3.7.2 Proof of Theorem 7.1.2.- 3.7.3 r-function associated with ?n.- 3.8 Symmetries of the Garnier system 𝒢nand of the system ?n.- 3.8.1 Symmetries of 𝒢n.- 3.8.2 Symmetries of ?n.- 3.8.3 Prolongation of the system ?n.- 3.9 Particular solutions of the Hamiltonian system ?n.- 3.9.1 The Lauricella hypergeometric series FD.- 3.9.2 Particular solutions of ?n.- 4. Studies on Singularities of Non-linear Differential Equations.- 4.1 Singularities of regular type.- 4.1.1 Holomorphic solutions.- 4.1.2 One-dimensional case.- 4.1.2.1 Formal transformations.- 4.1.2.2 Convergence of formal transformations.- 4.1.3 The n-dimensional case.- 4.2 Fixed singular points of regular type of Painlevé equations.- 4.2.1 Transformation into the normal form.- 4.2.2 Solutions of equations in normal form.- 4.2.3 Proof of Theorem 2.2.1.- 4.2.4 Solutions of Painlevé equations.- Notes on the chapter titlepage illustrations.- Index of symbols.