9783528063559
From Gauss To Painlevé. A Modern Theory Of Special Functions
Vieweg (1991)
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#2775

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Functions, Special

Gamma function, the zeta function, the theta function, the hypergeometric 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 irregirregular 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 funcfunctions. 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 for 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 dimendimensional 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.

Although many students and researchers in mathematics and in physics need to learn about Painleve functions, it is not easy for them, even for professional mathematicians not specialized in the theory, to attain the heart of these functions. Indeed, up to now no systematic presentation of the subject was ever written in a western language. In order to fill the gap, the present book was devised as an introductory exposition to the Painleve functions using only plain language from a modern mathematical view point.

Chapter 1 states elementary theorems in the theory of differential equations needed in the sequel. Most basic theorems are stated without proof. In Chapter 2, the Gauss hypergeometric differential equation is thoroughly studied from various angles. This is not so much because our presentation is intended to be elementary but especially because this difdifferential equation provides the whole theory of linear differential equaequations with a leading example. In Chapter 3, the monodromy-preserving deformations of second-order Fuchsian equations are presented and the Hamiltonian structure of the so-called Gamier system is exhibited. The simplest Gaxnier system turns out to be the celebrated sixth Painleve equation, of which algebraic properties axe explained. Furthermore, it is shown that for some particular values of the parameters, the Garnier system admits solutions which are expressible by the Appell-Lauricella hypergeometric function. Finally in Chapter 4, the behavior of solutions of non-linear differential equations at singular points is investigated. A modern treatment of classical theories is given and a new method to attack the singularities of the Painleve equations is introduced.

In what follows, only elementary notions of differential equations (the existence theorem for ordinary differential equations, etc.), of funcfunction theory (power series, the Caiichy theorem, residues, etc.) and of group theory (normal subgroups, quotient groups, etc.) are supposed to be known, so that any graduate student should be able to understand the presentation.

Product Details
LoC Classification QA351 .F9 1991
Dewey 515/.5
Format Hardcover
Cover Price 56,00 €
No. of Pages 347
Height x Width 230 mm
Personal Details
Links Library of Congress