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**Linear differential equations and group theory from Riemann to Poincaré.**
*(English)*
Zbl 0596.01018

Boston-Basel-Stuttgart: Birkhäuser. XXV, 460 p. DM 118.00 (1986).

The volume under review is an ambitious undertaking with a dual purpose, first, to describe the development of the topics named in the title from an historical perspective; second, to expound the mathematics itself for the edification of the ambitious undergraduate or graduate student (or working mathematician) who would like to know about it. The author’s command of the subject matter and skill as a writer are such that both goals are achieved admirably. The book contains an amazing wealth of material relating to the algebra, geometry, and analysis of the nineteenth century. Although the main focus is on the group-theoretic and geometric aspects of differential equations, the appearance of Abelian integrals and such associated topics as theta functions is inevitable in any account of the subject. Taking as his point of departure Gauss’ 1812 papers on the hypergeometric series, the author uses this series as the natural unifying principle to introduce subsequent work by Jacobi, Kummer and Riemann. The first chapter closes with a digression on some questions of rigor related to the techniques used to study differential equations at the time. Chapter 2 is devoted to the work of L. Fuchs, a speciality of the author [see his article in Bull. Am. Math. Soc., New. Ser. 10, 1- 26 (1984; Zbl 0536.01013)], and concludes with some of the work of Fuchs’ successors. Group theory enters the story in chapter 3, where the problem of finding algebraic solutions of the hypergeometric equation is discussed and the solutions obtained by Schwarz, Fuchs, Klein, Gordan, and Jordan are described. Chapter 4 treats modular equations through the work of Hermite, Fuchs, and Dedekind and describes the Galois theory of modular equations as developed by Betti, Hermite, Kronecker, and Klein. Chapter 5 describes various papers on algebraic curves and manages to squeeze in a brief account of the Jacobi inversion problem and the theory of theta functions. Finally chapter 6 discusses automorphic functions and brings the account down to the Poincaré-Klein collaboration of the early 1880’s. The volume concludes with a series of appendices, notes, and an index.

The author’s secondary purpose of publishing a sort of belated textbook of nineteenth-century analysis is no impediment to the carefully- annotated historical writing. Indeed, without the author’s lucid explanations it would be very difficult for a novice to orient himself in the historical material, with its unavoidable technical details. Only occasionally does the reader wonder whether a result that is a consequence of previous work was actually recognized as such in the period under discussion. The reviewer can imagine no more enjoyable way to learn or revisit this material. Written with accurate historical perspective and clear exposition, this book is truly hard to put down.

The author’s secondary purpose of publishing a sort of belated textbook of nineteenth-century analysis is no impediment to the carefully- annotated historical writing. Indeed, without the author’s lucid explanations it would be very difficult for a novice to orient himself in the historical material, with its unavoidable technical details. Only occasionally does the reader wonder whether a result that is a consequence of previous work was actually recognized as such in the period under discussion. The reviewer can imagine no more enjoyable way to learn or revisit this material. Written with accurate historical perspective and clear exposition, this book is truly hard to put down.

Reviewer: R.L.Cooke

### MSC:

01A55 | History of mathematics in the 19th century |

01-02 | Research exposition (monographs, survey articles) pertaining to history and biography |

34-03 | History of ordinary differential equations |

20-03 | History of group theory |