A First Course on Complex Functions

Author: G. Jameson
Publisher: Springer Science & Business Media
ISBN: 9789400956803
Release Date: 2013-03-09
Genre: Science

This book contains a rigorous coverage of those topics (and only those topics) that, in the author's judgement, are suitable for inclusion in a first course on Complex Functions. Roughly speaking, these can be summarized as being the things that can be done with Cauchy's integral formula and the residue theorem. On the theoretical side, this includes the basic core of the theory of differentiable complex functions, a theory which is unsurpassed in Mathematics for its cohesion, elegance and wealth of surprises. On the practical side, it includes the computational applications of the residue theorem. Some prominence is given to the latter, because for the more sceptical student they provide the justification for inventing the complex numbers. Analytic continuation and Riemann surfaces form an essentially different chapter of Complex Analysis. A proper treatment is far too sophisticated for a first course, and they are therefore excluded. The aim has been to produce the simplest possible rigorous treatment of the topics discussed. For the programme outlined above, it is quite sufficient to prove Cauchy'S integral theorem for paths in star-shaped open sets, so this is done. No form of the Jordan curve theorem is used anywhere in the book.

Real and Complex Analysis

Author: Christopher Apelian
Publisher: CRC Press
ISBN: 1584888075
Release Date: 2009-12-08
Genre: Mathematics

Presents Real & Complex Analysis Together Using a Unified Approach A two-semester course in analysis at the advanced undergraduate or first-year graduate level Unlike other undergraduate-level texts, Real and Complex Analysis develops both the real and complex theory together. It takes a unified, elegant approach to the theory that is consistent with the recommendations of the MAA’s 2004 Curriculum Guide. By presenting real and complex analysis together, the authors illustrate the connections and differences between these two branches of analysis right from the beginning. This combined development also allows for a more streamlined approach to real and complex function theory. Enhanced by more than 1,000 exercises, the text covers all the essential topics usually found in separate treatments of real analysis and complex analysis. Ancillary materials are available on the book’s website. This book offers a unique, comprehensive presentation of both real and complex analysis. Consequently, students will no longer have to use two separate textbooks—one for real function theory and one for complex function theory.

Complex Variables

Author: Steven G. Krantz
Publisher: CRC Press
ISBN: 9781420010954
Release Date: 2007-09-19
Genre: Mathematics

From the algebraic properties of a complete number field, to the analytic properties imposed by the Cauchy integral formula, to the geometric qualities originating from conformality, Complex Variables: A Physical Approach with Applications and MATLAB explores all facets of this subject, with particular emphasis on using theory in practice. The first five chapters encompass the core material of the book. These chapters cover fundamental concepts, holomorphic and harmonic functions, Cauchy theory and its applications, and isolated singularities. Subsequent chapters discuss the argument principle, geometric theory, and conformal mapping, followed by a more advanced discussion of harmonic functions. The author also presents a detailed glimpse of how complex variables are used in the real world, with chapters on Fourier and Laplace transforms as well as partial differential equations and boundary value problems. The final chapter explores computer tools, including Mathematica®, MapleTM, and MATLAB®, that can be employed to study complex variables. Each chapter contains physical applications drawing from the areas of physics and engineering. Offering new directions for further learning, this text provides modern students with a powerful toolkit for future work in the mathematical sciences.

Algebraic Numbers and Algebraic Functions

Author: P.M. Cohn
Publisher: CRC Press
ISBN: 0412361906
Release Date: 1991-09-01
Genre: Mathematics

This book is an introduction to the theory of algebraic numbers and algebraic functions of one variable. The basic development is the same for both using E Artin's legant approach, via valuations. Number Theory is pursued as far as the unit theorem and the finiteness of the class number. In function theory the aim is the Abel-Jacobi theorem describing the devisor class group, with occasional geometrical asides to help understanding. Assuming only an undergraduate course in algebra, plus a little acquaintance with topology and complex function theory, the book serves as an introduction to more technical works in algebraic number theory, function theory or algebraic geometry by an exposition of the central themes in the subject.

Option Valuation

Author: Hugo D. Junghenn
Publisher: CRC Press
ISBN: 9781439889114
Release Date: 2011-11-23
Genre: Business & Economics

Option Valuation: A First Course in Financial Mathematics provides a straightforward introduction to the mathematics and models used in the valuation of financial derivatives. It examines the principles of option pricing in detail via standard binomial and stochastic calculus models. Developing the requisite mathematical background as needed, the text presents an introduction to probability theory and stochastic calculus suitable for undergraduate students in mathematics, economics, and finance. The first nine chapters of the book describe option valuation techniques in discrete time, focusing on the binomial model. The author shows how the binomial model offers a practical method for pricing options using relatively elementary mathematical tools. The binomial model also enables a clear, concrete exposition of fundamental principles of finance, such as arbitrage and hedging, without the distraction of complex mathematical constructs. The remaining chapters illustrate the theory in continuous time, with an emphasis on the more mathematically sophisticated Black-Scholes-Merton model. Largely self-contained, this classroom-tested text offers a sound introduction to applied probability through a mathematical finance perspective. Numerous examples and exercises help students gain expertise with financial calculus methods and increase their general mathematical sophistication. The exercises range from routine applications to spreadsheet projects to the pricing of a variety of complex financial instruments. Hints and solutions to odd-numbered problems are given in an appendix and a full solutions manual is available for qualifying instructors.

A Concise Introduction to Pure Mathematics Second Edition

Author: Martin Liebeck
Publisher: CRC Press
ISBN: 1584885475
Release Date: 2005-11-02
Genre: Mathematics

A Concise Introduction to Pure Mathematics, Second Edition provides a robust bridge between high school and university mathematics, expanding upon basic topics in ways that will interest first-year students in mathematics and related fields and stimulate further study. Divided into 22 short chapters, this textbook offers a selection of exercises ranging from routine calculations to quite challenging problems. The author discusses real and complex numbers and explains how these concepts are applied in solving natural problems. He introduces topics in analysis, geometry, number theory, and combinatorics. What's New in the Second Edition: Contains extra material concerning prime numbers, forming the basis for data encryption Explores "Secret Codes" - one of today's most spectacular applications of pure mathematics Discusses Permutations and their importance in many topics in discrete mathematics The textbook allows for the design of courses with various points of emphasis, because it can be divided into four fairly independent sections related to: an introduction to number systems and analysis; theory of the integers; an introduction to discrete mathematics; and functions, relations, and countability.

Mathematical and Algorithmic Foundations of the Internet

Author: Fabrizio Luccio
Publisher: CRC Press
ISBN: 9781439831380
Release Date: 2011-07-06
Genre: Computers

To truly understand how the Internet and Web are organized and function requires knowledge of mathematics and computation theory. Mathematical and Algorithmic Foundations of the Internet introduces the concepts and methods upon which computer networks rely and explores their applications to the Internet and Web. The book offers a unique approach to mathematical and algorithmic concepts, demonstrating their universality by presenting ideas and examples from various fields, including literature, history, and art. Progressing from fundamental concepts to more specific topics and applications, the text covers computational complexity and randomness, networks and graphs, parallel and distributed computing, and search engines. While the mathematical treatment is rigorous, it is presented at a level that can be grasped by readers with an elementary mathematical background. The authors also present a lighter side to this complex subject by illustrating how many of the mathematical concepts have counterparts in everyday life. The book provides in-depth coverage of the mathematical prerequisites and assembles a complete presentation of how computer networks function. It is a useful resource for anyone interested in the inner functioning, design, and organization of the Internet.

COMPLEX VARIABLES

Author: H. S. KASANA
Publisher: PHI Learning Pvt. Ltd.
ISBN: 8120326415
Release Date: 2005-01-01
Genre: Mathematics

The second edition of this comprehensive and accessible text continues to offer students a challenging and enjoyable study of complex variables that is infused with perfect balanced coverage of mathematical theory and applied topics. The author explains fundamental concepts and techniques with precision and introduces the students to complex variable theory through conceptual develop-ment of analysis that enables them to develop a thorough understanding of the topics discussed. Geometric interpretation of the results, wherever necessary, has been inducted for making the analysis more accessible. The level of the text assumes that the reader is acquainted with elementary real analysis. Beginning with the revision of the algebra of complex variables, the book moves on to deal with analytic functions, elementary functions, complex integration, sequences, series and infinite products, series expansions, singularities and residues. The application-oriented chapters on sums and integrals, conformal mappings, Laplace transform, and some special topics, provide a practical-use perspective. Enriched with many numerical examples and exercises designed to test the student's comprehension of the topics covered, this book is written for a one-semester course in complex variables for students in the science and engineering disciplines.

Measure and Integral

Author: Richard L. Wheeden
Publisher: CRC Press
ISBN: 9781498702904
Release Date: 2015-04-24
Genre: Mathematics

Now considered a classic text on the topic, Measure and Integral: An Introduction to Real Analysis provides an introduction to real analysis by first developing the theory of measure and integration in the simple setting of Euclidean space, and then presenting a more general treatment based on abstract notions characterized by axioms and with less geometric content. Published nearly forty years after the first edition, this long-awaited Second Edition also: Studies the Fourier transform of functions in the spaces L1, L2, and Lp, 1 p Shows the Hilbert transform to be a bounded operator on L2, as an application of the L2 theory of the Fourier transform in the one-dimensional case Covers fractional integration and some topics related to mean oscillation properties of functions, such as the classes of Hölder continuous functions and the space of functions of bounded mean oscillation Derives a subrepresentation formula, which in higher dimensions plays a role roughly similar to the one played by the fundamental theorem of calculus in one dimension Extends the subrepresentation formula derived for smooth functions to functions with a weak gradient Applies the norm estimates derived for fractional integral operators to obtain local and global first-order Poincaré–Sobolev inequalities, including endpoint cases Proves the existence of a tangent plane to the graph of a Lipschitz function of several variables Includes many new exercises not present in the first edition This widely used and highly respected text for upper-division undergraduate and first-year graduate students of mathematics, statistics, probability, or engineering is revised for a new generation of students and instructors. The book also serves as a handy reference for professional mathematicians.

Mathematics

Author: Keith J. Devlin
Publisher: Columbia University Press
ISBN: 023111639X
Release Date: 1999
Genre: Mathematics

A modern classic by an accomplished mathematician and best-selling author has been updated to encompass and explain the recent headline-making advances in the field in non-technical terms.

Mathematical Programming and Control Theory

Author: B. D. Craven
Publisher: Springer Science & Business Media
ISBN: 9789400957961
Release Date: 2012-12-06
Genre: Science

In a mathematical programming problem, an optimum (maxi mum or minimum) of a function is sought, subject to con straints on the values of the variables. In the quarter century since G. B. Dantzig introduced the simplex method for linear programming, many real-world problems have been modelled in mathematical programming terms. Such problems often arise in economic planning - such as scheduling industrial production or transportation - but various other problems, such as the optimal control of an interplanetary rocket, are of similar kind. Often the problems involve nonlinear func tions, and so need methods more general than linear pro gramming. This book presents a unified theory of nonlinear mathe matical programming. The same methods and concepts apply equally to 'nonlinear programming' problems with a finite number of variables, and to 'optimal control' problems with e. g. a continuous curve (i. e. infinitely many variables). The underlying ideas of vector space, convex cone, and separating hyperplane are the same, whether the dimension is finite or infinite; and infinite dimension makes very little difference to the proofs. Duality theory - the various nonlinear generaliz ations of the well-known duality theorem of linear program ming - is found relevant also to optimal control, and the , PREFACE Pontryagin theory for optimal control also illuminates finite dimensional problems. The theory is simplified, and its applicability extended, by using the geometric concept of convex cones, in place of coordinate inequalities.

Introduction to Optimization Methods

Author: P. Adby
Publisher: Springer Science & Business Media
ISBN: 9789400957053
Release Date: 2013-03-09
Genre: Science

During the last decade the techniques of non-linear optim ization have emerged as an important subject for study and research. The increasingly widespread application of optim ization has been stimulated by the availability of digital computers, and the necessity of using them in the investigation of large systems. This book is an introduction to non-linear methods of optimization and is suitable for undergraduate and post graduate courses in mathematics, the physical and social sciences, and engineering. The first half of the book covers the basic optimization techniques including linear search methods, steepest descent, least squares, and the Newton-Raphson method. These are described in detail, with worked numerical examples, since they form the basis from which advanced methods are derived. Since 1965 advanced methods of unconstrained and constrained optimization have been developed to utilise the computational power of the digital computer. The second half of the book describes fully important algorithms in current use such as variable metric methods for unconstrained problems and penalty function methods for constrained problems. Recent work, much of which has not yet been widely applied, is reviewed and compared with currently popular techniques under a few generic main headings. vi PREFACE Chapter I describes the optimization problem in mathemat ical form and defines the terminology used in the remainder of the book. Chapter 2 is concerned with single variable optimization. The main algorithms of both search and approximation methods are developed in detail since they are an essential part of many multi-variable methods.

Sets Functions and Logic

Author: Keith Devlin
Publisher: CRC Press
ISBN: 0203498739
Release Date: 2003-11-24
Genre: Mathematics

Keith Devlin. You know him. You've read his columns in MAA Online, you've heard him on the radio, and you've seen his popular mathematics books. In between all those activities and his own research, he's been hard at work revising Sets, Functions and Logic, his standard-setting text that has smoothed the road to pure mathematics for legions of undergraduate students. Now in its third edition, Devlin has fully reworked the book to reflect a new generation. The narrative is more lively and less textbook-like. Remarks and asides link the topics presented to the real world of students' experience. The chapter on complex numbers and the discussion of formal symbolic logic are gone in favor of more exercises, and a new introductory chapter on the nature of mathematics--one that motivates readers and sets the stage for the challenges that lie ahead. Students crossing the bridge from calculus to higher mathematics need and deserve all the help they can get. Sets, Functions, and Logic, Third Edition is an affordable little book that all of your transition-course students not only can afford, but will actually read...and enjoy...and learn from. About the Author Dr. Keith Devlin is Executive Director of Stanford University's Center for the Study of Language and Information and a Consulting Professor of Mathematics at Stanford. He has written 23 books, one interactive book on CD-ROM, and over 70 published research articles. He is a Fellow of the American Association for the Advancement of Science, a World Economic Forum Fellow, and a former member of the Mathematical Sciences Education Board of the National Academy of Sciences,. Dr. Devlin is also one of the world's leading popularizers of mathematics. Known as "The Math Guy" on NPR's Weekend Edition, he is a frequent contributor to other local and national radio and TV shows in the US and Britain, writes a monthly column for the Web journal MAA Online, and regularly writes on mathematics and computers for the British newspaper The Guardian.

Solution Techniques for Elementary Partial Differential Equations Third Edition

Author: Christian Constanda
Publisher: CRC Press
ISBN: 9781498704960
Release Date: 2016-03-30
Genre: Mathematics

Solution Techniques for Elementary Partial Differential Equations, Third Edition remains a top choice for a standard, undergraduate-level course on partial differential equations (PDEs). Making the text even more user-friendly, this third edition covers important and widely used methods for solving PDEs. New to the Third Edition New sections on the series expansion of more general functions, other problems of general second-order linear equations, vibrating string with other types of boundary conditions, and equilibrium temperature in an infinite strip Reorganized sections that make it easier for students and professors to navigate the contents Rearranged exercises that are now at the end of each section/subsection instead of at the end of the chapter New and improved exercises and worked examples A brief Mathematica® program for nearly all of the worked examples, showing students how to verify results by computer This bestselling, highly praised textbook uses a streamlined, direct approach to develop students’ competence in solving PDEs. It offers concise, easily understood explanations and worked examples that allow students to see the techniques in action.

An Introduction to Systems Biology

Author: Uri Alon
Publisher: CRC Press
ISBN: 9781584886426
Release Date: 2006-07-07
Genre: Mathematics

Thorough and accessible, this book presents the design principles of biological systems, and highlights the recurring circuit elements that make up biological networks. It provides a simple mathematical framework which can be used to understand and even design biological circuits. The textavoids specialist terms, focusing instead on several well-studied biological systems that concisely demonstrate key principles. An Introduction to Systems Biology: Design Principles of Biological Circuits builds a solid foundation for the intuitive understanding of general principles. It encourages the reader to ask why a system is designed in a particular way and then proceeds to answer with simplified models.