Complex Analysis 2: Riemann Surfaces, Several Complex Variables, Abelian Functions, Higher Modular Functions

ISBN-13: 9783642205538 / Angielski / Miękka / 2011 / 506 str.

Eberhard Freitag

Opis

The book contains a complete self-contained introduction to highlights of classical complex analysis. New proofs and some new results are included. All needed notions are developed within the book: with the exception of some basic facts which can be found in the rst volume. There is no comparable treatment in the literature.

Recenzje

From the reviews:

"The book under review is the second volume of the textbook Complex analysis, consisting of 8 chapters. It provides an approach to the theory of Riemann surfaces from complex analysis. ... The book is self-contained and, moreover, some notions which might be unfamiliar for the reader are explained in appendices of chapters. ... this book is an excellent textbook on Riemann surfaces, especially for graduate students who have taken the first course of complex analysis." (Hiroshige Shiga, Mathematical Reviews, Issue 2012 f)

"The book under review is largely self-contained, pleasantly down-to-earth, remarkably versatile, and highly educating simultaneously. No doubt, this fine textbook provides an excellent source for the further study of more advanced and topical themes in the theory of Riemann surfaces, their Jacobians and moduli spaces, and in the general theory of complex Abelian varieties and modular forms likewise. It is very welcome that the English translation of the German original has been made available so quickly!" (Werner Kleinert, Zentralblatt MATH, Vol. 1234, 2012)

"The author provides a (very brief) introduction to fundamental notions of topology, but develops fully the theory of surfaces and covering spaces he needs. ... the book includes a proof of the classification of compact orientable surfaces by their genus. ... this one is definitely a graduate text. ... There is a lot of mathematics in this book, presented efficiently and well. ... It is a book I am glad to have, and that I will certainly refer to in the future." (Fernando Q. Gouvêa, The Mathematical Association of America, May, 2012)

Spis treści

Chapter I. Riemann Surfaces.- Chapter II. Harmonic Functions on Riemann Surfaces.- Chapter III. Uniformization.- Chapter IV. Compact Riemann Surfaces.- Appendices to Chapter IV.- Chapter V. Analytic Functions of Several Complex Variables.- Chapter V. Analytic Functions of Several Complex Variable.- Chapter VI. Abelian Functions.- Chapter VII. Modular Forms of Several Variables.- Chapter VIII. Appendix: Algebraic Tools.- References.- Index.

O autorze

Prof. Dr. Eberhard Freitag, Universität Heidelberg, Mathematisches Institut

Blurb

The book provides a complete presentation of complex analysis, starting with the theory of Riemann surfaces, including uniformization theory and a detailed treatment of the theory of compact Riemann surfaces, the Riemann-Roch theorem, Abel's theorem and Jacobi's inversion theorem. This motivates a short introduction into the theory of several complex variables, followed by the theory of Abelian functions up to the theta theorem. The last part of the book provides an introduction into the theory of higher modular functions.