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**Mathematical analysis III. Analytic functions, differentials and varieties, Riemann surfaces.
(Analyse mathématique III. Fonctions analytiques, différentielles et variétés, surfaces de Riemann.)**
*(French)*
Zbl 0987.30001

Berlin: Springer. ix, 338 p. (2002).

This is the third volume of the course of mathematical analysis teached by Roger Godement at the University of Paris. The first two volumes have already been analyzed (Zbl 0908.26001 and Zbl 0908.26002), and the third one is written in the same spirit. Besides the technical aspects, written in a careful and luminous style, the reader will find many historical and personal remarks, including, on p. 148-154, a defense of the role of Bourbaki in reply of some remarks of B. Mandelbrot, and, on p. 201-202, a comparison of the “proofs” of the Stokes theorem by physicists and mathematicians.

The first topics treated in this volume is Cauchy’s theory of holomorphic functions, including a very careful treatment of the integral theorems, and detailed applications to the real and complex Fourier transforms, gamma function, Hankel integral, Mellin transform and Dirichlet problem on a half-plane. The next chapter develops the calculus of functions of several variables, tensor calculus, differential forms, differential manifolds and Stokes theorem. It includes a detailed treatment of the formula of change of variables in a multiple integral. The last chapter is devoted to a detailed treatment of the Riemann surface of an algebraic function. It ends with a very vivid description of the algebraic viewpoint.

The pleasure and interest in reading such a book is the same as for the first two volumes, and one is looking forward discovering the fourth and last one.

The first topics treated in this volume is Cauchy’s theory of holomorphic functions, including a very careful treatment of the integral theorems, and detailed applications to the real and complex Fourier transforms, gamma function, Hankel integral, Mellin transform and Dirichlet problem on a half-plane. The next chapter develops the calculus of functions of several variables, tensor calculus, differential forms, differential manifolds and Stokes theorem. It includes a detailed treatment of the formula of change of variables in a multiple integral. The last chapter is devoted to a detailed treatment of the Riemann surface of an algebraic function. It ends with a very vivid description of the algebraic viewpoint.

The pleasure and interest in reading such a book is the same as for the first two volumes, and one is looking forward discovering the fourth and last one.

Reviewer: J.Mawhin (Louvain-La-Neuve)