Complex Functions

An Algebraic and Geometric Viewpoint

Author: Gareth A. Jones,David Singerman

Publisher: Cambridge University Press

ISBN: 9780521313667

Category: Mathematics

Page: 342

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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 Möbius 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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Functions of Several Complex Variables and Their Singularities

Author: Wolfgang Ebeling

Publisher: American Mathematical Soc.

ISBN: 0821833197

Category: Mathematics

Page: 312

View: 5356

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The book provides an introduction to the theory of functions of several complex variables and their singularities, with special emphasis on topological aspects. The topics include Riemann surfaces, holomorphic functions of several variables, classification and deformation of singularities, fundamentals of differential topology, and the topology of singularities. The aim of the book is to guide the reader from the fundamentals to more advanced topics of recent research. All the necessary prerequisites are specified and carefully explained. The general theory is illustrated by various examples and applications.
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Geometry of the Quintic

Author: Jerry Shurman

Publisher: John Wiley & Sons

ISBN: 9780471130178

Category: Mathematics

Page: 200

View: 3260

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A chance for students to apply a wide range of mathematics to an engaging problem This book helps students at the advanced undergraduate and beginning graduate levels to develop connections between the algebra, geometry, and analysis that they know, and to better appreciate the totality of what they have learned. The text demonstrates the use of general concepts by applying theorems from various areas in the context of one problem––solving the quintic. The problem is approached from two directions: the first is Felix Klein′s nineteenth–century approach, using the icosahedron. The second approach presents recent works of Peter Doyle and Curt McMullen, which update Klein′s use of transcendental functions to a solution through pure iteration. Filling a pedagogical gap in the literature and providing a solid platform from which to address more advanced material, this meticulously written book: ∗ Develops the Riemann sphere and its field of functions, classifies the finite groups of its automorphisms, computes for each such group a generator of the group–invariant functions, and discusses algebraic aspects of inverting this generator ∗ Gives, in the case of the icosahedral group, an elegant presentation of the relevant icosahedral geometry and its relation to the Brioschi quintic ∗ Reduces the general quintic to Brioschi form by radicals ∗ Proves Kronecker′s theorem that an "auxiliary" square root is necessary for any such reduction ∗ Expounds Doyle and McMullen′s development of an iterative solution to the quintic ∗ Provides a wealth of exercises and illustrations to clarify the geometry of the quintic
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An Introduction to Teichmüller Spaces

Author: Yoichi Imayoshi,Masahiko Taniguchi

Publisher: Springer Science & Business Media

ISBN: 4431681744

Category: Mathematics

Page: 279

View: 8777

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This book offers an easy and compact access to the theory of Teichmüller spaces, starting from the most elementary aspects to the most recent developments, e.g. the role this theory plays with regard to string theory. Teichmüller spaces give parametrization of all the complex structures on a given Riemann surface. This subject is related to many different areas of mathematics including complex analysis, algebraic geometry, differential geometry, topology in two and three dimensions, Kleinian and Fuchsian groups, automorphic forms, complex dynamics, and ergodic theory. Recently, Teichmüller spaces have begun to play an important role in string theory. Imayoshi and Taniguchi have attempted to make the book as self-contained as possible. They present numerous examples and heuristic arguments in order to help the reader grasp the ideas of Teichmüller theory. The book will be an excellent source of information for graduate students and reserachers in complex analysis and algebraic geometry as well as for theoretical physicists working in quantum theory.
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Complex Analysis

Author: Eberhard Freitag,Rolf Busam

Publisher: Taylor & Francis

ISBN: 9783540257240

Category: Mathematics

Page: 547

View: 8490

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The guiding principle of this presentation of ``Classical Complex Analysis'' is to proceed as quickly as possible to the central results while using a small number of notions and concepts from other fields. Thus the prerequisites for understanding this book are minimal; only elementary facts of calculus and algebra are required. The first four chapters cover the essential core of complex analysis: - differentiation in C (including elementary facts about conformal mappings) - integration in C (including complex line integrals, Cauchy's Integral Theorem, and the Integral Formulas) - sequences and series of analytic functions, (isolated) singularities, Laurent series, calculus of residues - construction of analytic functions: the gamma function, Weierstrass' Factorization Theorem, Mittag-Leffler Partial Fraction Decomposition, and -as a particular highlight- the Riemann Mapping Theorem, which characterizes the simply connected domains in C. Further topics included are: - the theory of elliptic functions based on the model of K. Weierstrass (with an excursions to older approaches due to N.H. Abel and C.G.J. Jacobi using theta series) - an introduction to the theory of elliptic modular functions and elliptic modular forms - the use of complex analysis to obtain number theoretical results - a proof of the Prime Number Theorem with a weak form of the error term. The book is especially suited for graduated students in mathematics and advanced undergraduated students in mathematics and other sciences. Motivating introductions, more than four hundred exercises of all levels of difficulty with hints or solutions, historical annotations, and over 120 figures make the overall presentation very attractive. The structure of the text, including abstracts beginning each chapter and highlighting of the main results, makes this book very appropriate for self-guided study and an indispensable aid in preparing for tests. This English edition is based on the fourth forthcoming German edition.
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An Introduction to Algebraic Geometry

Author: Kenji Ueno

Publisher: American Mathematical Soc.

ISBN: 0821811444

Category: Mathematics

Page: 246

View: 522

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This introduction to algebraic geometry allows readers to grasp the fundamentals of the subject with only linear algebra and calculus as prerequisites. After a brief history of the subject, the book introduces projective spaces and projective varieties, and explains plane curves and resolution of their singularities. The volume further develops the geometry of algebraic curves and treats congruence zeta functions of algebraic curves over a finite field. It concludes with a complex analytical discussion of algebraic curves. The author emphasizes computation of concrete examples rather than proofs, and these examples are discussed from various viewpoints. This approach allows readers to develop a deeper understanding of the theorems.
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Choice

Publication of the Association of College and Research Libraries, a Division of the American Library Association

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Academic libraries

Page: N.A

View: 8821

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