Valiron G. The Geometric Theory of Ordinary Differential Equations and Algebraic Functions

Math Sci Press, 1984. - 576 pages.The main purpose of this volume is to expose the student to the material of a course in differential and integral calculus, which was not covered in the volume entitled Theorie des functions, namely: the elements of the theory of analytic functions of several variables, the theory of differential equations and first order partial differential equations, the calculus of variations and applications of analysis to geometry. But, as in the first volume, I have made some inclusions to the theories studied, occasionally with some detail. As certain results relating to the properties of solutions of analytic differential equations necessitate some acquaintance with the theory of algehraic functions, I have devoted two chapters to this theory and that of abelian integrals. This has allowed me to prove some theorems relating to algebraic plane curves, which are no longer taught in specialised mathematics courses, and to present, once more, the unity of mathematics.
Three types of equations defining functions are studied in this book: algebraic equations in two variables, (ordinary) differential equations and partial differential equations. All of these equations are functional equations, hence the justification in calling the first part of the title: Equations functionelles. As for applications, these are the applications of analysis to geometry and include some applications of Abel's theorem on abelian integrals.
The first two chapters are devoted to algebraic functions of one variable and to abelian integrals: the theorem of Picard on uniformisation is established with the help of a recent theorem due to Bloch. The elements of the theory of analytic functions of several variables and the method, due to Cauchy, of the majorant functions, are discussed in the third chapter: the theorems of existence of the solutions of analytic differential equations are established along with the essential inclusion due to Weierstrass and Poincare', concerning the way in which the solutions depend on the initial conditions and how the parameters arise in these equations. There follows a chapter on the theory of contact and (curve) envelopes; for the latter, I have strived to present an accurate account. This chapter immediately serves to produce an application of the theory of analytic functions, and a pathway to the study of singular solutions of differential equations.