An Introduction to Complex Function Theory (Undergraduate Texts in Mathematics)

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9781461269670: An Introduction to Complex Function Theory (Undergraduate Texts in Mathematics)

This book provides a rigorous yet elementary introduction to the theory of analytic functions of a single complex variable. While presupposing in its readership a degree of mathematical maturity, it insists on no formal prerequisites beyond a sound knowledge of calculus. Starting from basic definitions, the text slowly and carefully develops the ideas of complex analysis to the point where such landmarks of the subject as Cauchy's theorem, the Riemann mapping theorem, and the theorem of Mittag-Leffler can be treated without sidestepping any issues of rigor. The emphasis throughout is a geometric one, most pronounced in the extensive chapter dealing with conformal mapping, which amounts essentially to a "short course" in that important area of complex function theory. Each chapter concludes with a wide selection of exercises, ranging from straightforward computations to problems of a more conceptual and thought-provoking nature.

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Book Description Book Condition: New. Publisher/Verlag: Springer, Berlin | This book provides a rigorous yet elementary introduction to the theory of analytic functions of a single complex variable. While presupposing in its readership a degree of mathematical maturity, it insists on no formal prerequisites beyond a sound knowledge of calculus. Starting from basic definitions, the text slowly and carefully develops the ideas of complex analysis to the point where such landmarks of the subject as Cauchy's theorem, the Riemann mapping theorem, and the theorem of Mittag-Leffler can be treated without sidestepping any issues of rigor. The emphasis throughout is a geometric one, most pronounced in the extensive chapter dealing with conformal mapping, which amounts essentially to a "short course" in that important area of complex function theory. Each chapter concludes with a wide selection of exercises, ranging from straightforward computations to problems of a more conceptual and thought-provoking nature. | 'I The Complex Number System.- 1 The Algebra and Geometry of Complex Numbers.- 1.1 The Field of Complex Numbers.- 1.2 Conjugate, Modulus, and Argument.- 2 Exponentials and Logarithms of Complex Numbers.- 2.1 Raising e to Complex Powers.- 2.2 Logarithms of Complex Numbers.- 2.3 Raising Complex Numbers to Complex Powers.- 3 Functions of a Complex Variable.- 3.1 Complex Functions.- 3.2 Combining Functions.- 3.3 Functions as Mappings.- 4 Exercises for Chapter I.- II The Rudiments of Plane Topology.- 1 Basic Notation and Terminology.- 1.1 Disks.- 1.2 Interior Points, Open Sets.- 1.3 Closed Sets.- 1.4 Boundary, Closure, Interior.- 1.5 Sequences.- 1.6 Convergence of Complex Sequences.- 1.7 Accumulation Points of Complex Sequences.- 2 Continuity and Limits of Functions.- 2.1 Continuity.- 2.2 Limits of Functions.- 3 Connected Sets.- 3.1 Disconnected Sets.- 3.2 Connected Sets.- 3.3 Domains.- 3.4 Components of Open Sets.- 4 Compact Sets.- 4.1 Bounded Sets and Sequences.- 4.2 Cauchy Sequences.- 4.3 Compact Sets.- 4.4 Uniform Continuity.- 5 Exercises for Chapter II.- III Analytic Functions.- 1 Complex Derivatives.- 1.1 Differentiability.- 1.2 Differentiation Rules.- 1.3 Analytic Functions.- 2 The Cauchy-Riemann Equations.- 2.1 The Cauchy-Riemann System of Equations.- 2.2 Consequences of the Cauchy-Riemann Relations.- 3 Exponential and Trigonometric Functions.- 3.1 Entire Functions.- 3.2 Trigonometric Functions.- 3.3 The Principal Arcsine and Arctangent Functions.- 4 Branches of Inverse Functions.- 4.1 Branches of Inverse Functions.- 4.2 Branches of the pth-root Function.- 4.3 Branches of the Logarithm Function.- 4.4 Branches of the ?-power Function.- 5 Differentiability in the Real Sense.- 5.1 Real Differentiability.- 5.2 The Functions fz and fz.- 6 Exercises for Chapter III.- IV Complex Integration.- 1 Paths in the Complex Plane.- 1.1 Paths.- 1.2 Smooth and Piece wise Smooth Paths.- 1.3 Parametrizing Line Segments.- 1.4 Reverse Paths, Path Sums.- 1.5 Change of Parameter.- 2 Integrals Along Paths.- 2.1 Complex Line Integrals.- 2.2 Properties of Contour Integrals.- 2.3 Primitives.- 2.4 Some Notation.- 3 Rectiflable Paths.- 3.1 Rectifiable Paths.- 3.2 Integrals Along Rectifiable Paths.- 4 Exercises for Chapter IV.- V Cauchy's Theorem and its Consequences.- 1 The Local Cauchy Theorem.- 1.1 Cauchy's Theorem For Rectangles.- 1.2 Integrals and Primitives.- 1.3 The Local Cauchy Theorem.- 2 Winding Numbers and the Local Cauchy Integral Formula.- 2.1 Winding Numbers.- 2.2 Oriented Paths, Jordan Contours.- 2.3 The Local Integral Formula.- 3 Consequences of the Local Cauchy Integral Formula.- 3.1 Analyticity of Derivatives.- 3.2 Derivative Estimates.- 3.3 The Maximum Principle.- 4 More About Logarithm and Power Functions.- 4.1 Branches of Logarithms of Functions.- 4.2 Logarithms of Rational Functions.- 4.3 Branches of Powers of Functions.- 5 The Global Cauchy Theorems.- 5.1 Iterated Line Integrals.- 5.2 Cycles.- 5.3 Cauchy's Theorem and Integral Formula.- 6 Simply. Bookseller Inventory # K9781461269670

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Book Description Springer Sep 2012, 2012. Taschenbuch. Book Condition: Neu. Neuware - This book provides a rigorous yet elementary introduction to the theory of analytic functions of a single complex variable. While presupposing in its readership a degree of mathematical maturity, it insists on no formal prerequisites beyond a sound knowledge of calculus. Starting from basic definitions, the text slowly and carefully develops the ideas of complex analysis to the point where such landmarks of the subject as Cauchy's theorem, the Riemann mapping theorem, and the theorem of Mittag-Leffler can be treated without sidestepping any issues of rigor. The emphasis throughout is a geometric one, most pronounced in the extensive chapter dealing with conformal mapping, which amounts essentially to a 'short course' in that important area of complex function theory. Each chapter concludes with a wide selection of exercises, ranging from straightforward computations to problems of a more conceptual and thought-provoking nature. 584 pp. Englisch. Bookseller Inventory # 9781461269670

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Book Description Springer Sep 2012, 2012. Taschenbuch. Book Condition: Neu. Neuware - This book provides a rigorous yet elementary introduction to the theory of analytic functions of a single complex variable. While presupposing in its readership a degree of mathematical maturity, it insists on no formal prerequisites beyond a sound knowledge of calculus. Starting from basic definitions, the text slowly and carefully develops the ideas of complex analysis to the point where such landmarks of the subject as Cauchy's theorem, the Riemann mapping theorem, and the theorem of Mittag-Leffler can be treated without sidestepping any issues of rigor. The emphasis throughout is a geometric one, most pronounced in the extensive chapter dealing with conformal mapping, which amounts essentially to a 'short course' in that important area of complex function theory. Each chapter concludes with a wide selection of exercises, ranging from straightforward computations to problems of a more conceptual and thought-provoking nature. 584 pp. Englisch. Bookseller Inventory # 9781461269670

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Book Description Springer-Verlag New York Inc., United States, 2012. Paperback. Book Condition: New. Language: English . Brand New Book. This book provides a rigorous yet elementary introduction to the theory of analytic functions of a single complex variable. While presupposing in its readership a degree of mathematical maturity, it insists on no formal prerequisites beyond a sound knowledge of calculus. Starting from basic definitions, the text slowly and carefully develops the ideas of complex analysis to the point where such landmarks of the subject as Cauchy s theorem, the Riemann mapping theorem, and the theorem of Mittag-Leffler can be treated without sidestepping any issues of rigor. The emphasis throughout is a geometric one, most pronounced in the extensive chapter dealing with conformal mapping, which amounts essentially to a short course in that important area of complex function theory. Each chapter concludes with a wide selection of exercises, ranging from straightforward computations to problems of a more conceptual and thought-provoking nature. Softcover reprint of the original 1st ed. 1991. Bookseller Inventory # LIO9781461269670

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Book Description Springer-Verlag New York Inc., United States, 2012. Paperback. Book Condition: New. Language: English . Brand New Book. This book provides a rigorous yet elementary introduction to the theory of analytic functions of a single complex variable. While presupposing in its readership a degree of mathematical maturity, it insists on no formal prerequisites beyond a sound knowledge of calculus. Starting from basic definitions, the text slowly and carefully develops the ideas of complex analysis to the point where such landmarks of the subject as Cauchy s theorem, the Riemann mapping theorem, and the theorem of Mittag-Leffler can be treated without sidestepping any issues of rigor. The emphasis throughout is a geometric one, most pronounced in the extensive chapter dealing with conformal mapping, which amounts essentially to a short course in that important area of complex function theory. Each chapter concludes with a wide selection of exercises, ranging from straightforward computations to problems of a more conceptual and thought-provoking nature. Softcover reprint of the original 1st ed. 1991. Bookseller Inventory # LIO9781461269670

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Book Description Springer-Verlag New York Inc., United States, 2012. Paperback. Book Condition: New. Language: English . This book usually ship within 10-15 business days and we will endeavor to dispatch orders quicker than this where possible. Brand New Book. This book provides a rigorous yet elementary introduction to the theory of analytic functions of a single complex variable. While presupposing in its readership a degree of mathematical maturity, it insists on no formal prerequisites beyond a sound knowledge of calculus. Starting from basic definitions, the text slowly and carefully develops the ideas of complex analysis to the point where such landmarks of the subject as Cauchy s theorem, the Riemann mapping theorem, and the theorem of Mittag-Leffler can be treated without sidestepping any issues of rigor. The emphasis throughout is a geometric one, most pronounced in the extensive chapter dealing with conformal mapping, which amounts essentially to a short course in that important area of complex function theory. Each chapter concludes with a wide selection of exercises, ranging from straightforward computations to problems of a more conceptual and thought-provoking nature. Softcover reprint of the original 1st ed. 1991. Bookseller Inventory # LIE9781461269670

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Book Description Springer Sep 2012, 2012. Taschenbuch. Book Condition: Neu. This item is printed on demand - Print on Demand Neuware - This book provides a rigorous yet elementary introduction to the theory of analytic functions of a single complex variable. While presupposing in its readership a degree of mathematical maturity, it insists on no formal prerequisites beyond a sound knowledge of calculus. Starting from basic definitions, the text slowly and carefully develops the ideas of complex analysis to the point where such landmarks of the subject as Cauchy's theorem, the Riemann mapping theorem, and the theorem of Mittag-Leffler can be treated without sidestepping any issues of rigor. The emphasis throughout is a geometric one, most pronounced in the extensive chapter dealing with conformal mapping, which amounts essentially to a 'short course' in that important area of complex function theory. Each chapter concludes with a wide selection of exercises, ranging from straightforward computations to problems of a more conceptual and thought-provoking nature. 584 pp. Englisch. Bookseller Inventory # 9781461269670

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Book Description Springer-Verlag New York Inc., 2012. PAP. Book Condition: New. New Book. Shipped from US within 10 to 14 business days. THIS BOOK IS PRINTED ON DEMAND. Established seller since 2000. Bookseller Inventory # IQ-9781461269670

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