Explorations in Complex Functions
Explorations in Complex Functions

Author: Richard Beals

Publisher: Springer Nature

Published: 2020-10-19

Total Pages: 353

ISBN-13: 3030545334

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This textbook explores a selection of topics in complex analysis. From core material in the mainstream of complex analysis itself, to tools that are widely used in other areas of mathematics, this versatile compilation offers a selection of many different paths. Readers interested in complex analysis will appreciate the unique combination of topics and connections collected in this book. Beginning with a review of the main tools of complex analysis, harmonic analysis, and functional analysis, the authors go on to present multiple different, self-contained avenues to proceed. Chapters on linear fractional transformations, harmonic functions, and elliptic functions offer pathways to hyperbolic geometry, automorphic functions, and an intuitive introduction to the Schwarzian derivative. The gamma, beta, and zeta functions lead into L-functions, while a chapter on entire functions opens pathways to the Riemann hypothesis and Nevanlinna theory. Cauchy transforms give rise to Hilbert and Fourier transforms, with an emphasis on the connection to complex analysis. Valuable additional topics include Riemann surfaces, steepest descent, tauberian theorems, and the Wiener–Hopf method. Showcasing an array of accessible excursions, Explorations in Complex Functions is an ideal companion for graduate students and researchers in analysis and number theory. Instructors will appreciate the many options for constructing a second course in complex analysis that builds on a first course prerequisite; exercises complement the results throughout.

Explorations in Complex Analysis
Explorations in Complex Analysis

Author: Michael A. Brilleslyper

Publisher: American Mathematical Soc.

Published: 2012-12-31

Total Pages: 373

ISBN-13: 1614441081

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Research topics in the book include complex dynamics, minimal surfaces, fluid flows, harmonic, conformal, and polygonal mappings, and discrete complex analysis via circle packing. The nature of this book is different from many mathematics texts: the focus is on student-driven and technology-enhanced investigation. Interlaced in the reading for each chapter are examples, exercises, explorations, and projects, nearly all linked explicitly with computer applets for visualization and hands-on manipulation.

Function Theory of Several Complex Variables
Function Theory of Several Complex Variables

Author: Steven George Krantz

Publisher: American Mathematical Soc.

Published: 2001

Total Pages: 586

ISBN-13: 0821827243

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Emphasizing integral formulas, the geometric theory of pseudoconvexity, estimates, partial differential equations, approximation theory, inner functions, invariant metrics, and mapping theory, this title is intended for the student with a background in real and complex variable theory, harmonic analysis, and differential equations.

More Explorations in Complex Functions
More Explorations in Complex Functions

Author: Richard Beals

Publisher: Springer Nature

Published: 2023-07-01

Total Pages: 410

ISBN-13: 3031282884

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More Explorations in Complex Functions is something of a sequel to GTM 287, Explorations in Complex Functions. Both texts introduce a variety of topics, from core material in the mainstream of complex analysis to tools that are widely used in other areas of mathematics and applications, but there is minimal overlap between the two books. The intended readership is the same, namely graduate students and researchers in complex analysis, independent readers, seminar attendees, or instructors for a second course in complex analysis. Instructors will appreciate the many options for constructing a second course that builds on a standard first course in complex analysis. Exercises complement the results throughout. There is more material in this present text than one could expect to cover in a year’s course in complex analysis. A mapping of dependence relations among chapters enables instructors and independent readers a choice of pathway to reading the text. Chapters 2, 4, 5, 7, and 8 contain the function theory background for some stochastic equations of current interest, such as SLE. The text begins with two introductory chapters to be used as a resource. Chapters 3 and 4 are stand-alone introductions to complex dynamics and to univalent function theory, including deBrange’s theorem, respectively. Chapters 5—7 may be treated as a unit that leads from harmonic functions to covering surfaces to the uniformization theorem and Fuchsian groups. Chapter 8 is a stand-alone treatment of quasiconformal mapping that paves the way for Chapter 9, an introduction to Teichmüller theory. The final chapters, 10–14, are largely stand-alone introductions to topics of both theoretical and applied interest: the Bergman kernel, theta functions and Jacobi inversion, Padé approximants and continued fractions, the Riemann—Hilbert problem and integral equations, and Darboux’s method for computing asymptotics.

Complex Analysis
Complex Analysis

Author: Theodore W. Gamelin

Publisher: Springer Science & Business Media

Published: 2013-11-01

Total Pages: 508

ISBN-13: 0387216073

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An introduction to complex analysis for students with some knowledge of complex numbers from high school. It contains sixteen chapters, the first eleven of which are aimed at an upper division undergraduate audience. The remaining five chapters are designed to complete the coverage of all background necessary for passing PhD qualifying exams in complex analysis. Topics studied include Julia sets and the Mandelbrot set, Dirichlet series and the prime number theorem, and the uniformization theorem for Riemann surfaces, with emphasis placed on the three geometries: spherical, euclidean, and hyperbolic. Throughout, exercises range from the very simple to the challenging. The book is based on lectures given by the author at several universities, including UCLA, Brown University, La Plata, Buenos Aires, and the Universidad Autonomo de Valencia, Spain.

Methods of the Theory of Functions of Many Complex Variables
Methods of the Theory of Functions of Many Complex Variables

Author: Vasiliy Sergeyevich Vladimirov

Publisher: Courier Corporation

Published: 2007-01-01

Total Pages: 370

ISBN-13: 0486458121

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This systematic exposition outlines the fundamentals of the theory of single sheeted domains of holomorphy. It further illustrates applications to quantum field theory, the theory of functions, and differential equations with constant coefficients. Students of quantum field theory will find this text of particular value. The text begins with an introduction that defines the basic concepts and elementary propositions, along with the more salient facts from the theory of functions of real variables and the theory of generalized functions. Subsequent chapters address the theory of plurisubharmonic functions and pseudoconvex domains, along with characteristics of domains of holomorphy. These explorations are further examined in terms of four types of domains: multiple-circular, tubular, semitubular, and Hartogs' domains. Surveys of integral representations focus on the Martinelli-Bochner, Bergman-Weil, and Bochner representations. The final chapter is devoted to applications, particularly those involved in field theory. It employs the theory of generalized functions, along with the theory of functions of several complex variables.

Coloring Book of Complex Function Representations
Coloring Book of Complex Function Representations

Author: Julie Barnes

Publisher: MAA Press

Published: 2017-12-31

Total Pages: 40

ISBN-13: 9780883855911

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Have you ever seen a mathematical object that was so intriguing that anyone, regardless of mathematical background, could appreciate its aesthetic beauty? If so, did you stop to color it? Now is your chance. Explore the beauty of mathematics in this collection if intricate pictures related to complex-valued functions. Any of these images could pass for designs found in some of the many coloring books for adults seen in stores today and are displayed here for you to color. While looking at the coloring pages, you can read about the author's quest to find interesting images. Their tale includes expeditions on the complex plane, work with families of complex functions, visits to Julia sets, unexpected results from a typo, random explorations, and a final send-off from a well-known cartoon character. Grab your colored pencils and enjoy coloring these functions. There are no incorrect ways to color, and consequently, there are no answers in the back of the book!

Explorations in Mathematical Physics
Explorations in Mathematical Physics

Author: Don Koks

Publisher: Springer Science & Business Media

Published: 2006-09-15

Total Pages: 549

ISBN-13: 0387309438

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Have you ever wondered why the language of modern physics centres on geometry? Or how quantum operators and Dirac brackets work? What a convolution really is? What tensors are all about? Or what field theory and lagrangians are, and why gravity is described as curvature? This book takes you on a tour of the main ideas forming the language of modern mathematical physics. Here you will meet novel approaches to concepts such as determinants and geometry, wave function evolution, statistics, signal processing, and three-dimensional rotations. You will see how the accelerated frames of special relativity tell us about gravity. On the journey, you will discover how tensor notation relates to vector calculus, how differential geometry is built on intuitive concepts, and how variational calculus leads to field theory. You will meet quantum measurement theory, along with Green functions and the art of complex integration, and finally general relativity and cosmology. The book takes a fresh approach to tensor analysis built solely on the metric and vectors, with no need for one-forms. This gives a much more geometrical and intuitive insight into vector and tensor calculus, together with general relativity, than do traditional, more abstract methods. Don Koks is a physicist at the Defence Science and Technology Organisation in Adelaide, Australia. His doctorate in quantum cosmology was obtained from the Department of Physics and Mathematical Physics at Adelaide University. Prior work at the University of Auckland specialised in applied accelerator physics, along with pure and applied mathematics.

The Calculus of Complex Functions
The Calculus of Complex Functions

Author: William Johnston

Publisher: American Mathematical Society

Published: 2022-04-01

Total Pages: 456

ISBN-13: 1470465655

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The book introduces complex analysis as a natural extension of the calculus of real-valued functions. The mechanism for doing so is the extension theorem, which states that any real analytic function extends to an analytic function defined in a region of the complex plane. The connection to real functions and calculus is then natural. The introduction to analytic functions feels intuitive and their fundamental properties are covered quickly. As a result, the book allows a surprisingly large coverage of the classical analysis topics of analytic and meromorphic functions, harmonic functions, contour integrals and series representations, conformal maps, and the Dirichlet problem. It also introduces several more advanced notions, including the Riemann hypothesis and operator theory, in a manner accessible to undergraduates. The last chapter describes bounded linear operators on Hilbert and Banach spaces, including the spectral theory of compact operators, in a way that also provides an excellent review of important topics in linear algebra and provides a pathway to undergraduate research topics in analysis. The book allows flexible use in a single semester, full-year, or capstone course in complex analysis. Prerequisites can range from only multivariate calculus to a transition course or to linear algebra or real analysis. There are over one thousand exercises of a variety of types and levels. Every chapter contains an essay describing a part of the history of the subject and at least one connected collection of exercises that together comprise a project-level exploration.

Classical Topics in Complex Function Theory
Classical Topics in Complex Function Theory

Author: Reinhold Remmert

Publisher: Springer Science & Business Media

Published: 2013-03-14

Total Pages: 362

ISBN-13: 1475729561

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An ideal text for an advanced course in the theory of complex functions, this book leads readers to experience function theory personally and to participate in the work of the creative mathematician. The author includes numerous glimpses of the function theory of several complex variables, which illustrate how autonomous this discipline has become. In addition to standard topics, readers will find Eisenstein's proof of Euler's product formula for the sine function; Wielandts uniqueness theorem for the gamma function; Stirlings formula; Isssas theorem; Besses proof that all domains in C are domains of holomorphy; Wedderburns lemma and the ideal theory of rings of holomorphic functions; Estermanns proofs of the overconvergence theorem and Blochs theorem; a holomorphic imbedding of the unit disc in C3; and Gausss expert opinion on Riemanns dissertation. Remmert elegantly presents the material in short clear sections, with compact proofs and historical comments interwoven throughout the text. The abundance of examples, exercises, and historical remarks, as well as the extensive bibliography, combine to make an invaluable source for students and teachers alike