TY  - GEN
AU  - Godement,Roger
TI  - Analysis III : : analytic and differential functions, manifolds and Riemann surfaces 
T2  - Universitext  
SN  - 9783319160528
U1  - 515 23
PY  - 2015///
CY  - Switzerland
PB  - Springer
KW  - Mathematics
N1  - Includes index; vii. Cauchy theory--
ix. Multivariate differential and integral calculus--
x. The Riemann surface of an algebraic function--
Index--
Table of contents of volume I--
Table of contents of volume II
N2  - 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).
ER  -