Applications Of Holomorphic Functions In Geometry

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Bicomplex Holomorphic Functions

Author : M. Elena Luna-Elizarrarás
Publisher : Birkhäuser
ISBN 13 : 3319248685
Total Pages : 231 pages
Book Rating : 4.3/5 (192 download)

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Book Synopsis Bicomplex Holomorphic Functions by : M. Elena Luna-Elizarrarás

Download or read book Bicomplex Holomorphic Functions written by M. Elena Luna-Elizarrarás and published by Birkhäuser. This book was released on 2015-12-11 with total page 231 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of this book is to develop the foundations of the theory of holomorphicity on the ring of bicomplex numbers. Accordingly, the main focus is on expressing the similarities with, and differences from, the classical theory of one complex variable. The result is an elementary yet comprehensive introduction to the algebra, geometry and analysis of bicomplex numbers. Around the middle of the nineteenth century, several mathematicians (the best known being Sir William Hamilton and Arthur Cayley) became interested in studying number systems that extended the field of complex numbers. Hamilton famously introduced the quaternions, a skew field in real-dimension four, while almost simultaneously James Cockle introduced a commutative four-dimensional real algebra, which was rediscovered in 1892 by Corrado Segre, who referred to his elements as bicomplex numbers. The advantages of commutativity were accompanied by the introduction of zero divisors, something that for a while dampened interest in this subject. In recent years, due largely to the work of G.B. Price, there has been a resurgence of interest in the study of these numbers and, more importantly, in the study of functions defined on the ring of bicomplex numbers, which mimic the behavior of holomorphic functions of a complex variable. While the algebra of bicomplex numbers is a four-dimensional real algebra, it is useful to think of it as a “complexification” of the field of complex numbers; from this perspective, the bicomplex algebra possesses the properties of a one-dimensional theory inside four real dimensions. Its rich analysis and innovative geometry provide new ideas and potential applications in relativity and quantum mechanics alike. The book will appeal to researchers in the fields of complex, hypercomplex and functional analysis, as well as undergraduate and graduate students with an interest in one- or multidimensional complex analysis.

Application of Holomorphic Functions in Two and Higher Dimensions

Author : Klaus Gürlebeck
Publisher : Springer
ISBN 13 : 3034809646
Total Pages : 402 pages
Book Rating : 4.0/5 (348 download)

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Book Synopsis Application of Holomorphic Functions in Two and Higher Dimensions by : Klaus Gürlebeck

Download or read book Application of Holomorphic Functions in Two and Higher Dimensions written by Klaus Gürlebeck and published by Springer. This book was released on 2016-06-20 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents applications of hypercomplex analysis to boundary value and initial-boundary value problems from various areas of mathematical physics. Given that quaternion and Clifford analysis offer natural and intelligent ways to enter into higher dimensions, it starts with quaternion and Clifford versions of complex function theory including series expansions with Appell polynomials, as well as Taylor and Laurent series. Several necessary function spaces are introduced, and an operator calculus based on modifications of the Dirac, Cauchy-Fueter, and Teodorescu operators and different decompositions of quaternion Hilbert spaces are proved. Finally, hypercomplex Fourier transforms are studied in detail. All this is then applied to first-order partial differential equations such as the Maxwell equations, the Carleman-Bers-Vekua system, the Schrödinger equation, and the Beltrami equation. The higher-order equations start with Riccati-type equations. Further topics include spatial fluid flow problems, image and multi-channel processing, image diffusion, linear scale invariant filtering, and others. One of the highlights is the derivation of the three-dimensional Kolosov-Mushkelishvili formulas in linear elasticity. Throughout the book the authors endeavor to present historical references and important personalities. The book is intended for a wide audience in the mathematical and engineering sciences and is accessible to readers with a basic grasp of real, complex, and functional analysis.

From Holomorphic Functions to Complex Manifolds

Author : Klaus Fritzsche
Publisher : Springer Science & Business Media
ISBN 13 : 146849273X
Total Pages : 406 pages
Book Rating : 4.4/5 (684 download)

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Book Synopsis From Holomorphic Functions to Complex Manifolds by : Klaus Fritzsche

Download or read book From Holomorphic Functions to Complex Manifolds written by Klaus Fritzsche and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 406 pages. Available in PDF, EPUB and Kindle. Book excerpt: This introduction to the theory of complex manifolds covers the most important branches and methods in complex analysis of several variables while completely avoiding abstract concepts involving sheaves, coherence, and higher-dimensional cohomology. Only elementary methods such as power series, holomorphic vector bundles, and one-dimensional cocycles are used. Each chapter contains a variety of examples and exercises.

Holomorphic Functions and Integral Representations in Several Complex Variables

Author : R. Michael Range
Publisher : Springer Science & Business Media
ISBN 13 : 1475719183
Total Pages : 405 pages
Book Rating : 4.4/5 (757 download)

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Book Synopsis Holomorphic Functions and Integral Representations in Several Complex Variables by : R. Michael Range

Download or read book Holomorphic Functions and Integral Representations in Several Complex Variables written by R. Michael Range and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 405 pages. Available in PDF, EPUB and Kindle. Book excerpt: The subject of this book is Complex Analysis in Several Variables. This text begins at an elementary level with standard local results, followed by a thorough discussion of the various fundamental concepts of "complex convexity" related to the remarkable extension properties of holomorphic functions in more than one variable. It then continues with a comprehensive introduction to integral representations, and concludes with complete proofs of substantial global results on domains of holomorphy and on strictly pseudoconvex domains inC", including, for example, C. Fefferman's famous Mapping Theorem. The most important new feature of this book is the systematic inclusion of many of the developments of the last 20 years which centered around integral representations and estimates for the Cauchy-Riemann equations. In particu lar, integral representations are the principal tool used to develop the global theory, in contrast to many earlier books on the subject which involved methods from commutative algebra and sheaf theory, and/or partial differ ential equations. I believe that this approach offers several advantages: (1) it uses the several variable version of tools familiar to the analyst in one complex variable, and therefore helps to bridge the often perceived gap between com plex analysis in one and in several variables; (2) it leads quite directly to deep global results without introducing a lot of new machinery; and (3) concrete integral representations lend themselves to estimations, therefore opening the door to applications not accessible by the earlier methods.

Theory and Applications of Holomorphic Functions on Algebraic Varieties over Arbitrary Ground Fields

Author : Oscar Zariski
Publisher : American Mathematical Soc.
ISBN 13 : 082181205X
Total Pages : 99 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Theory and Applications of Holomorphic Functions on Algebraic Varieties over Arbitrary Ground Fields by : Oscar Zariski

Download or read book Theory and Applications of Holomorphic Functions on Algebraic Varieties over Arbitrary Ground Fields written by Oscar Zariski and published by American Mathematical Soc.. This book was released on 1951 with total page 99 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Complex Geometry

Author : Daniel Huybrechts
Publisher : Springer Science & Business Media
ISBN 13 : 9783540212904
Total Pages : 336 pages
Book Rating : 4.2/5 (129 download)

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Book Synopsis Complex Geometry by : Daniel Huybrechts

Download or read book Complex Geometry written by Daniel Huybrechts and published by Springer Science & Business Media. This book was released on 2005 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: Easily accessible Includes recent developments Assumes very little knowledge of differentiable manifolds and functional analysis Particular emphasis on topics related to mirror symmetry (SUSY, Kaehler-Einstein metrics, Tian-Todorov lemma)

Complex Analytic Sets

Author : E.M. Chirka
Publisher : Springer Science & Business Media
ISBN 13 : 940092366X
Total Pages : 386 pages
Book Rating : 4.4/5 (9 download)

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Book Synopsis Complex Analytic Sets by : E.M. Chirka

Download or read book Complex Analytic Sets written by E.M. Chirka and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 386 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of complex analytic sets is part of the modern geometrical theory of functions of several complex variables. A wide circle of problems in multidimensional complex analysis, related to holomorphic functions and maps, can be reformulated in terms of analytic sets. In these reformulations additional phenomena may emerge, while for the proofs new methods are necessary. (As an example we can mention the boundary properties of conformal maps of domains in the plane, which may be studied by means of the boundary properties of the graphs of such maps.) The theory of complex analytic sets is a relatively young branch of complex analysis. Basically, it was developed to fulfill the need of the theory of functions of several complex variables, but for a long time its development was, so to speak, within the framework of algebraic geometry - by analogy with algebraic sets. And although at present the basic methods of the theory of analytic sets are related with analysis and geometry, the foundations of the theory are expounded in the purely algebraic language of ideals in commutative algebras. In the present book I have tried to eliminate this noncorrespondence and to give a geometric exposition of the foundations of the theory of complex analytic sets, using only classical complex analysis and a minimum of algebra (well-known properties of polynomials of one variable). Moreover, it must of course be taken into consideration that algebraic geometry is one of the most important domains of application of the theory of analytic sets, and hence a lot of attention is given in the present book to algebraic sets.

The Convenient Setting of Global Analysis

Author : Andreas Kriegl
Publisher : American Mathematical Society
ISBN 13 : 1470478935
Total Pages : 631 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis The Convenient Setting of Global Analysis by : Andreas Kriegl

Download or read book The Convenient Setting of Global Analysis written by Andreas Kriegl and published by American Mathematical Society. This book was released on 2024-08-15 with total page 631 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book lays the foundations of differential calculus in infinite dimensions and discusses those applications in infinite dimensional differential geometry and global analysis not involving Sobolev completions and fixed point theory. The approach is simple: a mapping is called smooth if it maps smooth curves to smooth curves. Up to Fr‚chet spaces, this notion of smoothness coincides with all known reasonable concepts. In the same spirit, calculus of holomorphic mappings (including Hartogs' theorem and holomorphic uniform boundedness theorems) and calculus of real analytic mappings are developed. Existence of smooth partitions of unity, the foundations of manifold theory in infinite dimensions, the relation between tangent vectors and derivations, and differential forms are discussed thoroughly. Special emphasis is given to the notion of regular infinite dimensional Lie groups. Many applications of this theory are included: manifolds of smooth mappings, groups of diffeomorphisms, geodesics on spaces of Riemannian metrics, direct limit manifolds, perturbation theory of operators, and differentiability questions of infinite dimensional representations.

Geometry of Holomorphic Mappings

Author : Sergey Pinchuk
Publisher : Springer Nature
ISBN 13 : 3031371496
Total Pages : 217 pages
Book Rating : 4.0/5 (313 download)

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Book Synopsis Geometry of Holomorphic Mappings by : Sergey Pinchuk

Download or read book Geometry of Holomorphic Mappings written by Sergey Pinchuk and published by Springer Nature. This book was released on 2023-10-16 with total page 217 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph explores the problem of boundary regularity and analytic continuation of holomorphic mappings between domains in complex Euclidean spaces. Many important methods and techniques in several complex variables have been developed in connection with these questions, and the goal of this book is to introduce the reader to some of these approaches and to demonstrate how they can be used in the context of boundary properties of holomorphic maps. The authors present substantial results concerning holomorphic mappings in several complex variables with improved and often simplified proofs. Emphasis is placed on geometric methods, including the Kobayashi metric, the Scaling method, Segre varieties, and the Reflection principle. Geometry of Holomorphic Mappings will provide a valuable resource for PhD students in complex analysis and complex geometry; it will also be of interest to researchers in these areas as a reference.

Numerical Range of Holomorphic Mappings and Applications

Author : Mark Elin
Publisher : Springer
ISBN 13 : 3030050203
Total Pages : 238 pages
Book Rating : 4.0/5 (3 download)

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Book Synopsis Numerical Range of Holomorphic Mappings and Applications by : Mark Elin

Download or read book Numerical Range of Holomorphic Mappings and Applications written by Mark Elin and published by Springer. This book was released on 2019-03-11 with total page 238 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book describes recent developments as well as some classical results regarding holomorphic mappings. The book starts with a brief survey of the theory of semigroups of linear operators including the Hille-Yosida and the Lumer-Phillips theorems. The numerical range and the spectrum of closed densely defined linear operators are then discussed in more detail and an overview of ergodic theory is presented. The analytic extension of semigroups of linear operators is also discussed. The recent study of the numerical range of composition operators on the unit disk is mentioned. Then, the basic notions and facts in infinite dimensional holomorphy and hyperbolic geometry in Banach and Hilbert spaces are presented, L. A. Harris' theory of the numerical range of holomorphic mappings is generalized, and the main properties of the so-called quasi-dissipative mappings and their growth estimates are studied. In addition, geometric and quantitative analytic aspects of fixed point theory are discussed. A special chapter is devoted to applications of the numerical range to diverse geometric and analytic problems.

Differential Geometry and Its Applications

Author : Oldřich Kowalski
Publisher : World Scientific
ISBN 13 : 9812790616
Total Pages : 732 pages
Book Rating : 4.8/5 (127 download)

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Book Synopsis Differential Geometry and Its Applications by : Oldřich Kowalski

Download or read book Differential Geometry and Its Applications written by Oldřich Kowalski and published by World Scientific. This book was released on 2008 with total page 732 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains invited lectures and selected research papers in the fields of classical and modern differential geometry, global analysis, and geometric methods in physics, presented at the 10th International Conference on Differential Geometry and its Applications (DGA2007), held in Olomouc, Czech Republic.The book covers recent developments and the latest results in the following fields: Riemannian geometry, connections, jets, differential invariants, the calculus of variations on manifolds, differential equations, Finsler structures, and geometric methods in physics. It is also a celebration of the 300th anniversary of the birth of one of the greatest mathematicians, Leonhard Euler, and includes the Euler lecture OC Leonhard Euler OCo 300 years onOCO by R Wilson. Notable contributors include J F Cariena, M Castrilln Lpez, J Erichhorn, J-H Eschenburg, I KoliO, A P Kopylov, J Korbai, O Kowalski, B Kruglikov, D Krupka, O Krupkovi, R L(r)andre, Haizhong Li, S Maeda, M A Malakhaltsev, O I Mokhov, J Muoz Masqu(r), S Preston, V Rovenski, D J Saunders, M Sekizawa, J Slovik, J Szilasi, L Tamissy, P Walczak, and others."

Complex Analysis

Author : Rubí E. Rodríguez
Publisher : Springer Science & Business Media
ISBN 13 : 1441973230
Total Pages : 319 pages
Book Rating : 4.4/5 (419 download)

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Book Synopsis Complex Analysis by : Rubí E. Rodríguez

Download or read book Complex Analysis written by Rubí E. Rodríguez and published by Springer Science & Business Media. This book was released on 2012-11-28 with total page 319 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors’ aim here is to present a precise and concise treatment of those parts of complex analysis that should be familiar to every research mathematician. They follow a path in the tradition of Ahlfors and Bers by dedicating the book to a very precise goal: the statement and proof of the Fundamental Theorem for functions of one complex variable. They discuss the many equivalent ways of understanding the concept of analyticity, and offer a leisure exploration of interesting consequences and applications. Readers should have had undergraduate courses in advanced calculus, linear algebra, and some abstract algebra. No background in complex analysis is required.

Stein Manifolds and Holomorphic Mappings

Author : Franc Forstnerič
Publisher : Springer
ISBN 13 : 3319610589
Total Pages : 569 pages
Book Rating : 4.3/5 (196 download)

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Book Synopsis Stein Manifolds and Holomorphic Mappings by : Franc Forstnerič

Download or read book Stein Manifolds and Holomorphic Mappings written by Franc Forstnerič and published by Springer. This book was released on 2017-09-05 with total page 569 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book, now in a carefully revised second edition, provides an up-to-date account of Oka theory, including the classical Oka-Grauert theory and the wide array of applications to the geometry of Stein manifolds. Oka theory is the field of complex analysis dealing with global problems on Stein manifolds which admit analytic solutions in the absence of topological obstructions. The exposition in the present volume focuses on the notion of an Oka manifold introduced by the author in 2009. It explores connections with elliptic complex geometry initiated by Gromov in 1989, with the Andersén-Lempert theory of holomorphic automorphisms of complex Euclidean spaces and of Stein manifolds with the density property, and with topological methods such as homotopy theory and the Seiberg-Witten theory. Researchers and graduate students interested in the homotopy principle in complex analysis will find this book particularly useful. It is currently the only work that offers a comprehensive introduction to both the Oka theory and the theory of holomorphic automorphisms of complex Euclidean spaces and of other complex manifolds with large automorphism groups.

Rigid Analytic Geometry and Its Applications

Author : Jean Fresnel
Publisher : Springer Science & Business Media
ISBN 13 : 1461200415
Total Pages : 303 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Rigid Analytic Geometry and Its Applications by : Jean Fresnel

Download or read book Rigid Analytic Geometry and Its Applications written by Jean Fresnel and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 303 pages. Available in PDF, EPUB and Kindle. Book excerpt: Rigid (analytic) spaces were invented to describe degenerations, reductions, and moduli of algebraic curves and abelian varieties. This work, a revised and greatly expanded new English edition of an earlier French text by the same authors, presents important new developments and applications of the theory of rigid analytic spaces to abelian varieties, "points of rigid spaces," étale cohomology, Drinfeld modular curves, and Monsky-Washnitzer cohomology. The exposition is concise, self-contained, rich in examples and exercises, and will serve as an excellent graduate-level text for the classroom or for self-study.

Differential Geometry and Its Applications

Author : John Oprea
Publisher : American Mathematical Society
ISBN 13 : 1470477912
Total Pages : 497 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Differential Geometry and Its Applications by : John Oprea

Download or read book Differential Geometry and Its Applications written by John Oprea and published by American Mathematical Society. This book was released on 2024-07-01 with total page 497 pages. Available in PDF, EPUB and Kindle. Book excerpt: Differential Geometry and Its Applications studies the differential geometry of surfaces with the goal of helping students make the transition from the compartmentalized courses in a standard university curriculum to a type of mathematics that is a unified whole. It mixes geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences. That mix of ideas offers students the opportunity to visualize concepts through the use of computer algebra systems such as Maple. Differential Geometry and Its Applications emphasizes that this visualization goes hand in hand with understanding the mathematics behind the computer construction. The book is rich in results and exercises that form a continuous spectrum, from those that depend on calculation to proofs that are quite abstract.

Differential Geometry And Its Applications - Proceedings Of The 10th International Conference On Dga2007

Author : Demeter Krupka
Publisher : World Scientific
ISBN 13 : 9814471941
Total Pages : 732 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis Differential Geometry And Its Applications - Proceedings Of The 10th International Conference On Dga2007 by : Demeter Krupka

Download or read book Differential Geometry And Its Applications - Proceedings Of The 10th International Conference On Dga2007 written by Demeter Krupka and published by World Scientific. This book was released on 2008-07-14 with total page 732 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains invited lectures and selected research papers in the fields of classical and modern differential geometry, global analysis, and geometric methods in physics, presented at the 10th International Conference on Differential Geometry and its Applications (DGA2007), held in Olomouc, Czech Republic.The book covers recent developments and the latest results in the following fields: Riemannian geometry, connections, jets, differential invariants, the calculus of variations on manifolds, differential equations, Finsler structures, and geometric methods in physics. It is also a celebration of the 300th anniversary of the birth of one of the greatest mathematicians, Leonhard Euler, and includes the Euler lecture “Leonhard Euler — 300 years on” by R Wilson. Notable contributors include J F Cariñena, M Castrillón López, J Erichhorn, J-H Eschenburg, I Kolář, A P Kopylov, J Korbaš, O Kowalski, B Kruglikov, D Krupka, O Krupková, R Léandre, Haizhong Li, S Maeda, M A Malakhaltsev, O I Mokhov, J Muñoz Masqué, S Preston, V Rovenski, D J Saunders, M Sekizawa, J Slovák, J Szilasi, L Tamássy, P Walczak, and others.

A Course in Complex Analysis and Riemann Surfaces

Author : Wilhelm Schlag
Publisher : American Mathematical Society
ISBN 13 : 0821898477
Total Pages : 402 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis A Course in Complex Analysis and Riemann Surfaces by : Wilhelm Schlag

Download or read book A Course in Complex Analysis and Riemann Surfaces written by Wilhelm Schlag and published by American Mathematical Society. This book was released on 2014-08-06 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: Complex analysis is a cornerstone of mathematics, making it an essential element of any area of study in graduate mathematics. Schlag's treatment of the subject emphasizes the intuitive geometric underpinnings of elementary complex analysis that naturally lead to the theory of Riemann surfaces. The book begins with an exposition of the basic theory of holomorphic functions of one complex variable. The first two chapters constitute a fairly rapid, but comprehensive course in complex analysis. The third chapter is devoted to the study of harmonic functions on the disk and the half-plane, with an emphasis on the Dirichlet problem. Starting with the fourth chapter, the theory of Riemann surfaces is developed in some detail and with complete rigor. From the beginning, the geometric aspects are emphasized and classical topics such as elliptic functions and elliptic integrals are presented as illustrations of the abstract theory. The special role of compact Riemann surfaces is explained, and their connection with algebraic equations is established. The book concludes with three chapters devoted to three major results: the Hodge decomposition theorem, the Riemann-Roch theorem, and the uniformization theorem. These chapters present the core technical apparatus of Riemann surface theory at this level. This text is intended as a detailed, yet fast-paced intermediate introduction to those parts of the theory of one complex variable that seem most useful in other areas of mathematics, including geometric group theory, dynamics, algebraic geometry, number theory, and functional analysis. More than seventy figures serve to illustrate concepts and ideas, and the many problems at the end of each chapter give the reader ample opportunity for practice and independent study.