This chapter is devoted to some immediate consequences of the Fundamental Theorem for Cauchy Theo... more This chapter is devoted to some immediate consequences of the Fundamental Theorem for Cauchy Theory, Theorem 4.52, of the last chapter. Although the chapter is very short, it includes proofs of many of the implications of the Fundamental Theorem 1.1. We point out that these relatively compact proofs of a host of major theorems result from the work put in Chapter 4 and earlier chapters.
This chapter is devoted to an important method for constructing holomorphic functions. The tool i... more This chapter is devoted to an important method for constructing holomorphic functions. The tool is convergent power series. It is the basis for the introduction of new non-algebraic holomorphic functions, called elementary transcendental functions. It will turn out that all holomorphic functions are described (at least locally) by this tool. This will be proven in the next chapter.
Proceedings of the National Academy of Sciences of the United States of America, Jul 1, 1969
Let r be a (nonelementary) Kleinian group and q > 2 an integer. The group r acts in a natural way... more Let r be a (nonelementary) Kleinian group and q > 2 an integer. The group r acts in a natural way on the vector space II2g_2 of complex poly- nomials in one variable of degree < 2q -2. One can thus form HI(r,112q2), the first cohomology group of r with coefficients in l2q-2. There are essentially two ways of constructing cohomology classes. One construction originated with Eichler and has recently been extended by Ahlfors. Another construction is due to Bers. We show that for finitely generated r, every cohomology class pe H1(rAn2,2) can be written uniquely (if one chooses an invariant union of com- ponents of r) as a sum of a Bers cohomology class and an Eichler cohomology class. Similar decompositions are obtained for the subgroups of parabolic co- homology classes introduced by Alfors. Some information on the structure of
... D. students xxvii Lipman Bers, Complex Analyst by I. Kra and B. Maskit xxix Conformal dynamic... more ... D. students xxvii Lipman Bers, Complex Analyst by I. Kra and B. Maskit xxix Conformal dynamical systems and Lipman Bers by Frederick P. Gardiner and Linda Keen Ivii List of Publications Ixvii Research Papers 1. On bounded analytic functions of two complex variables in ...
This is one of the two most important chapters of this book. In it, we prove (based on the existe... more This is one of the two most important chapters of this book. In it, we prove (based on the existence theorems of the previous chapter) the three most important theorems concerning compact Riemann surfaces: the Riemann-Roch theorem, Abel’s theorem, and the Jacobi inversion theorem. Many applications of these theorems are obtained; and the simplest compact Riemann surfaces, the hyperelliptic ones, are discussed in great detail.
Library of Congress Cataloging in Publication Data Farkas, Hershel M Riemann surfaces. (Graduate ... more Library of Congress Cataloging in Publication Data Farkas, Hershel M Riemann surfaces. (Graduate texts in mathematics; v. 71) Bibliography: p. lncludes index. 1. Riemann surfaces. 1. Kra, Irwin, joint authoL II. Title. III. Series. QA333.F37 515'.223 79-24385 AII rights reserved. No part of this book may be translated or reprodueed in any form without written permission from Springer-Verlag.
This chapter is devoted to some immediate consequences of the Fundamental Theorem for Cauchy Theo... more This chapter is devoted to some immediate consequences of the Fundamental Theorem for Cauchy Theory, Theorem 4.52, of the last chapter. Although the chapter is very short, it includes proofs of many of the implications of the Fundamental Theorem 1.1. We point out that these relatively compact proofs of a host of major theorems result from the work put in Chapter 4 and earlier chapters.
This chapter is devoted to an important method for constructing holomorphic functions. The tool i... more This chapter is devoted to an important method for constructing holomorphic functions. The tool is convergent power series. It is the basis for the introduction of new non-algebraic holomorphic functions, called elementary transcendental functions. It will turn out that all holomorphic functions are described (at least locally) by this tool. This will be proven in the next chapter.
Proceedings of the National Academy of Sciences of the United States of America, Jul 1, 1969
Let r be a (nonelementary) Kleinian group and q > 2 an integer. The group r acts in a natural way... more Let r be a (nonelementary) Kleinian group and q > 2 an integer. The group r acts in a natural way on the vector space II2g_2 of complex poly- nomials in one variable of degree < 2q -2. One can thus form HI(r,112q2), the first cohomology group of r with coefficients in l2q-2. There are essentially two ways of constructing cohomology classes. One construction originated with Eichler and has recently been extended by Ahlfors. Another construction is due to Bers. We show that for finitely generated r, every cohomology class pe H1(rAn2,2) can be written uniquely (if one chooses an invariant union of com- ponents of r) as a sum of a Bers cohomology class and an Eichler cohomology class. Similar decompositions are obtained for the subgroups of parabolic co- homology classes introduced by Alfors. Some information on the structure of
... D. students xxvii Lipman Bers, Complex Analyst by I. Kra and B. Maskit xxix Conformal dynamic... more ... D. students xxvii Lipman Bers, Complex Analyst by I. Kra and B. Maskit xxix Conformal dynamical systems and Lipman Bers by Frederick P. Gardiner and Linda Keen Ivii List of Publications Ixvii Research Papers 1. On bounded analytic functions of two complex variables in ...
This is one of the two most important chapters of this book. In it, we prove (based on the existe... more This is one of the two most important chapters of this book. In it, we prove (based on the existence theorems of the previous chapter) the three most important theorems concerning compact Riemann surfaces: the Riemann-Roch theorem, Abel’s theorem, and the Jacobi inversion theorem. Many applications of these theorems are obtained; and the simplest compact Riemann surfaces, the hyperelliptic ones, are discussed in great detail.
Library of Congress Cataloging in Publication Data Farkas, Hershel M Riemann surfaces. (Graduate ... more Library of Congress Cataloging in Publication Data Farkas, Hershel M Riemann surfaces. (Graduate texts in mathematics; v. 71) Bibliography: p. lncludes index. 1. Riemann surfaces. 1. Kra, Irwin, joint authoL II. Title. III. Series. QA333.F37 515'.223 79-24385 AII rights reserved. No part of this book may be translated or reprodueed in any form without written permission from Springer-Verlag.
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