Analysis III electronic resource Analytic and Differential Functions, Manifolds and Riemann Surfaces / by Roger Godement.
Material type: TextSeries: UniversitextPublication details: Cham : Springer International Publishing : Imprint: Springer, 2015Description: VII, 321 p. 25 illus. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9783319160535Subject(s): mathematics | Functions of real variables | Mathematics | Real FunctionsDDC classification: 515.8 LOC classification: QA331.5Online resources: Click here to access onlineVIII Cauchy Theory -- IX Multivariate Differential and Integral Calculus -- X The Riemann Surface of an Algebraic Function.
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).
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