An introduction to complex analysis and geometry
- Responsibility
- John P. D'Angelo.
- 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
Math & Statistics Library
Stacks
| Call number | Status |
|---|---|
| QA331.7 .D356 2011 |
More options
Creators/Contributors
- Author/Creator
- D'Angelo, John P.
Contents/Summary
- Bibliography
- Includes bibliographical references (p. 159-160) 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 Cauchy-Riemann equations
- 4. Orthogonal trajectories and harmonic functions
- 5. A glimpse at harmonic functions
- 6. What is a differential form?
- ch. 6 Complex Integration
- 1. Complex-valued 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 minimum-maximum theorem
- 2. The fundamental theorem of algebra
- 3. Winding numbers, zeroes, and poles
- 4. Pythagorean triples
- 5. Elementary mappings
- 6. Quaternions
- 7. Higher-dimensional 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 Urbana-Champaign. 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)9780821852743 20160607
Bibliographic information
- Publication date
- 2010
- Series
- Pure and applied undergraduate texts ; 12
- The Sally series
- ISBN
- 9780821852743 (alk. paper)
- 0821852744 (alk. paper)
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