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Analysis III
Analytic and Differential Functions, Manifolds and Riemann Surfaces
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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.
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