An introduction to complex analysis and geometry
 Author/Creator
 D'Angelo, John P.
 Language
 English.
 Imprint
 Providence, R.I. : American Mathematical Society, 2010.
 Physical description
 xi, 163 p. ; 27 cm.
 Series

Pure and applied undergraduate texts ; 12.
Sally series (Providence, R.I.)
Access
Available online

Stacks

Unknown
QA331.7 .D356 2011

Unknown
QA331.7 .D356 2011
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Contents/Summary
 Bibliography
 Includes bibliographical references (p. 159160) and index.
 Contents

 Machine generated contents note: ch. 1 From the Real Numbers to the Complex Numbers
 1. Introduction
 2. Number systems
 3. Inequalities and ordered fields
 4. The complex numbers
 5. Alternative definitions of C
 6. A glimpse at metric spaces
 ch. 2 Complex Numbers
 1. Complex conjugation
 2. Existence of square roots
 3. Limits
 4. Convergent infinite series
 5. Uniform convergence and consequences
 6. The unit circle and trigonometry
 7. The geometry of addition and multiplication
 8. Logarithms
 ch. 3 Complex Numbers and Geometry
 1. Lines, circles, and balls
 2. Analytic geometry
 3. Quadratic polynomials
 4. Linear fractional transformations
 5. The Riemann sphere
 ch. 4 Power Series Expansions
 1. Geometric scries
 2. The radius of convergence
 3. Generating functions
 4. Fibonacci numbers
 5. An application of power series
 6. Rationality
 ch. 5 Complex Differentiation
 1. Definitions of complex analytic function
 2. Complex differentiation
 3. The CauchyRiemann equations
 4. Orthogonal trajectories and harmonic functions
 5. A glimpse at harmonic functions
 6. What is a differential form?
 ch. 6 Complex Integration
 1. Complexvalued functions
 2. Line integrals
 3. Goursat's proof
 4. The Cauchy integral formula
 5. A return to the definition of complex analytic function
 ch. 7 Applications of Complex Integration
 1. Singularities and residues
 2. Evaluating real integrals using complex variables methods
 3. Fourier transforms
 4. The Gamma function
 ch. 8 Additional Topics
 1. The minimummaximum theorem
 2. The fundamental theorem of algebra
 3. Winding numbers, zeroes, and poles
 4. Pythagorean triples
 5. Elementary mappings
 6. Quaternions
 7. Higherdimensional complex analysis.
 Publisher's Summary
 An Introduction to Complex Analysis and Geometry provides the reader with a deep appreciation of complex analysis and how this subject fits into mathematics. The book developed from courses given in the Campus Honors Program at the University of Illinois UrbanaChampaign. These courses aimed to share with students the way many mathematics and physics problems magically simplify when viewed from the perspective of complex analysis. The book begins at an elementary level but also contains advanced material.The first four chapters provide an introduction to complex analysis with many elementary and unusual applications. Chapters 5 through 7 develop the Cauchy theory and include some striking applications to calculus. Chapter 8 glimpses several appealing topics, simultaneously unifying the book and opening the door to further study.The 280 exercises range from simple computations to difficult problems. Their variety makes the book especially attractive.A reader of the first four chapters will be able to apply complex numbers in many elementary contexts. A reader of the full book will know basic one complex variable theory and will have seen it integrated into mathematics as a whole. Research mathematicians will discover several novel perspectives.
(source: Nielsen Book Data)
Bibliographic information
 Publication date
 2010
 Responsibility
 John P. D'Angelo.
 Series
 Pure and applied undergraduate texts ; 12
 The Sally series
 ISBN
 9780821852743
 0821852744