© 2015

Analysis III

Analytic and Differential Functions, Manifolds and Riemann Surfaces


  • Prefers ideas to calculations

  • Explains the ideas without parsimony of words

  • Based on 35 years of teaching at Paris University

  • Blends mathematics skillfully with didactical and historical considerations


Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-vii
  2. Roger Godement
    Pages 1-132
  3. Roger Godement
    Pages 275-309
  4. Back Matter
    Pages 311-321

About this book


Volume III sets out classical Cauchy theory. It is much more geared towards its innumerable applications than towards a more or less complete theory of analytic functions. Cauchy-type curvilinear integrals are then shown to generalize to any number of real variables (differential forms, Stokes-type formulas). The fundamentals of the theory of manifolds are then presented, mainly to provide the reader with a "canonical'' language and with some important theorems (change of variables in integration, differential equations). A final chapter shows how these theorems can be used to construct the compact Riemann surface of an algebraic function, a subject that is rarely addressed in the general literature though it only requires elementary techniques.

Besides the Lebesgue integral, Volume IV will set out a piece of specialized mathematics towards which the entire content of the previous volumes will converge: Jacobi, Riemann, Dedekind series and infinite products, elliptic functions, classical theory of modular functions and its modern version using the structure of the Lie algebra of SL(2,R).


Cauchy theory Riemann surface of an algebraic function complex Mellin and Fourier transforms differential forms and Stokes formula differential manifolds

Authors and affiliations

  1. 1.ParisFrance

About the authors

Roger Godement (October 1, 1921 - July 21, 2016) is known for his work in functional analysis and also his expository books. He started as a student at the École normale supérieure in 1940, where he became a student of Henri Cartan. He started research into harmonic analysis on locally compact abelian groups, finding a number of major results; this work was in parallel but independent of similar investigations in the USSR and Japan. Work on the abstract theory of spherical functions published in 1952 proved very influential in subsequent work, particularly that of Harish-Chandra. The isolation of the concept of square-integrable representation is attributed to him. The Godement compactness criterion in the theory of arithmetic groups was a conjecture of his. He later worked with Jacquet on the zeta function of a simple algebra. He was an active member of the Bourbaki group in the early 1950s, and subsequently gave a number of significant Bourbaki seminars. He also took part in the Cartan seminar. He also wrote texts on Lie groups, abstract algebra and mathematical analysis.

Bibliographic information

  • Book Title Analysis III
  • Book Subtitle Analytic and Differential Functions, Manifolds and Riemann Surfaces
  • Authors Roger Godement
  • Series Title Universitext
  • Series Abbreviated Title Universitext
  • DOI
  • Copyright Information Springer International Publishing Switzerland 2015
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-319-16052-8
  • eBook ISBN 978-3-319-16053-5
  • Series ISSN 0172-5939
  • Series E-ISSN 2191-6675
  • Edition Number 1
  • Number of Pages VII, 321
  • Number of Illustrations 25 b/w illustrations, 0 illustrations in colour
  • Topics Real Functions
  • Buy this book on publisher's site