de Gruyter Studies in Mathematics 28. Berlin: Walter de Gruyter (ISBN 3-11-017379-4/hbk). viii, 303 p. EUR 88.00 (2002).

Few books concentrating on topics in Painlevé equations have been published, especially in English, since the appearance of those equations in the beginning of the last century. For this reason, this book written by Gromak, Laine and Shimomura is valuable for those who wish to learn quickly the fundamental properties of Painlevé equations without the need to refer to a large number of original papers on them. In fact, this book, containing almost 300 pages and consisting of ten chapters, appendices, etc., deals exclusively with topics and materials, including some open problems, in the theory of Painlevé equations which range from past study to recent progress. In particular, it contains a detailed description of the complex analytic study of Painlevé transcendents, which is a distinguishing characteristic, because other books do not deal with this in detail. Most parts of this book are an exposition of the results obtained by or closely related to the three authors’ research on Painlevé equations.
The main part of this book begins with the study of the Painlevé property of the Painlevé equations (Chapter l). After the discovery of the Painlevé equations, Painlevé posed the question whether every solution of each Painlevé equation is free of movable singularities other than movable poles, i.e., whether each Painlevé equation has the so-called “Painlevé property” . He regarded this question as one of the most fundamental problems in the theory of Painlevé equations, and in fact, he tried to resolve affirmatively the question of the Painlevé property for the first Painlevé equation, but his proof of the Painlevé property had some gaps. The first rigorous proof of the Painlevé property for the first and sixth equations was given in 1960 by Hukuhara, who made precise estimates for the behavior of solutions. The idea of his proof is also applicable to proving the Painlevé property of the other Painlevé equations. In fact, several Japanese authors (Shimomura among them) gave proofs for those other equations.
Independently of the Japanese contributions, Hinkkanen-Laine and Steinmetz each gave other analytic proofs of the Painlevé property for Painlevé equations around the year 2000. Chapter 1 first gives a detailed proof of the Painlevé property for the first equation along the lines of Hukuhara and Shimomura, and next it gives a proof for the second equation based on Hinkkanen-Laine’s work using ideas coming from Painlevé’s attempt. It also gives a proof of the fourth equation by Steinmetz’ method of differential inequalities. Brief explanations are given regarding proofs for the other Painlevé equations. In every work mentioned here, the behavior of the Lyapunov functions corresponding to Painlevé equations are crucial to the proofs of the Painlevé property. Apart from the contributions mentioned above, one can find other proofs of the Painlevé property from the perspective of the theory of isomonodromic deformation of linear differential equations in Miwa’s and Malgrange’s works.
Chapter 2 deals with the growth of Painlevé transcendents. The study of the growth of Painlevé transcendents originated with Boutroux’ works. The first systematic treatment for it, by exploiting Nevanlinna theory, is due to a series of works by Wittich and Schubart in the 1950s and 1960s. Recently, the orders of growth for the Painlevé equations (except the sixth one) have been exactly and rigourously determined by Steinmetz and Shimomura independently. To determine the order of the growth $\rho(w)$ for the first transcendent $w$, the authors of this book first reproduce Steinmetz’ method of computing the characteristic function $T(r,w)$ for $w$, from which they conclude $\rho(w)\le\frac{5}{2}$ , and next prove the converse inequality $\rho(w)\ge \frac{5}{2}$ by using Shimomura’s lower estimate on $T(r,w)$. Moreover, following Hukuhara’s idea for the Painlevé property, the authors also determine the orders for the second and fourth transcendents. A brief description is given for the third and fifth transcendents.
Chapter 3, as well as Chapter 2, is based on Nevanlinna theory. The first half of this chapter deals with the evaluation of the deficiencies and the ramification indices of the Painlevé transcendents (except the sixth), where one can find some improvements by the authors of former studies in this field. The second half deals first with the reduction of Nevanlinna’s second main theorem for the Painlevé transcendents (except the sixth). Next, it deals with the evaluation of the deficiency $\delta(g,f)$ and the ramification index $\theta(g,f)$, where $f$ is the first or the second or the fourth transcendent {\it relative to a small function} $g$. These are generalized notions for the “normal” deficiency and the “normal” ramification index (for the definitions, see p. 72 in this book).
Apart from the complex analytic study of Painlevé transcendents, the successive six chapters, Chapters 4-9, deal with the comprehensive study of Painlevé equations comprised of contributions up to now by many people. Each of the six chapters are the exposition for each of the six Painlevé equations, respectively. The following subjects and materials for Painlevé equations and transcendents, some of which are under study even now, are the main focus of these chapters: Bäcklund transformations and symmetries of the equations; special solutions, especially rational or algebraic solutions and Riccati solutions; poles and the behavior at critical points of solutions; higher-order analogues of the equations; and the quotient by two entire functions representing a Painlevé transcendent. Based mainly on the studies of Belorussian mathematicians, including Gromak, poles and the behavior at critical points of Painlevé transcendents are discussed here in detail in connection with existence and construction of rational or algebraic solutions of the equations. Bäcklund transformations and symmetries are written in this book in the single equation forms for Painlevé equations, which are almost the same as the original forms adopted when such transformations were discovered mainly by Belorussian mathematicians, including Gromak, in the period from the 1950s to 1980s. As for special solutions, their existence conditions and constructions are discussed here in detail based on various studies up to now, and the description of them is given also in the single equation forms. Other systematic theories on transformations of Painlevé equations can be found in a series of works of Okamoto and Noumi-Yamada. As for a theoretical connection between special solutions and the irreducibility of Painlevé equations, Umemura’s works are recommended.
The last chapter, Chapter 10, makes some explanations on relationships between Painlevé equations and some nonlinear differenfcial equations appearing in mathematical physics, and gives a brief description on discrete Painlevé equations. In two appendices, the authors review some fundamental facts from the theory of complex differential equations and from Nevanlinna theory, which are frequently used in this book.
This book is regarded as an enlarged version of a previous one, [{\it V. I. Gromak} and {\it N. A. Lukashevich} [Analytic properties of solutions of Painlevé equations (Russian), Minsk: Izdatel’stvo Universitetskoe (1990;

Zbl 0752.34003)].