Complex Functions

Complex Functions : An Algebraic and Geometric Viewpoint

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Elliptic functions and Riemann surfaces played an important role in nineteenth-century mathematics. At the present time there is a great revival of interest in these topics not only for their own sake but also because of their applications to so many areas of mathematical research from group theory and number theory to topology and differential equations. In this book the authors give elementary accounts of many aspects of classical complex function theory including Mobius transformations, elliptic functions, Riemann surfaces, Fuchsian groups and modular functions. A distinctive feature of their presentation is the way in which they have incorporated into the text many interesting topics from other branches of mathematics. This book is based on lectures given to advanced undergraduates and is well-suited as a textbook for a second course in complex function theory. Professionals will also find it valuable as a straightforward introduction to a subject which is finding widespread application throughout mathematics.
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"...a very nice treatment which emphasizes the unity of mathematics...Several years ago the reviewer wanted to teach an undergraduate course that gave an introduction to hyperbolic geometry, Mobius transformations and discrete groups. There was no suitable undergraduate text. This book fills that void," Mathematical Reviews "Well motivated with good selection of problems at the end of each chapter." American Mathematical Monthly "...clear and well written...Its message, admirably conveyed, is that mathematics is not a collection of neat parcels, the contents of one being blind to the contents of another." Times Higher Education Supplement "...clearly presented...a delightful book..." American Scientist
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Table of contents

Introduction; 1. The Riemann sphere; 2. Mobius transformations; 3. Elliptic functions; 4. Meromorphic continuation and Riemann surfaces; 5. PSL(2,R) and its discrete subgroups; 6. The modular group; Appendices; References; Index of symbols; Index of names and definitions.
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