# Analysis III

## Analytic and Differential Functions, Manifolds and Riemann Surfaces

Part of the Universitext book series (UTX)

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Textbook

Part of the Universitext book series (UTX)

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

- DOI https://doi.org/10.1007/978-3-319-16053-5
- Copyright Information Springer International Publishing Switzerland 2015
- Publisher Name Springer, Cham
- eBook Packages Mathematics and Statistics
- Print ISBN 978-3-319-16052-8
- Online ISBN 978-3-319-16053-5
- Series Print ISSN 0172-5939
- Series Online ISSN 2191-6675
- Buy this book on publisher's site