An Introduction to Complex Analysis and Geometry (Pure and Applied Undergraduate Texts) (Pure and Applied Undergraduate Texts, 12)

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. Contents Preface Chapter 1. From the Real Numbers to the Complex Numbers 1. Introduction 2. Number systems 3. Inequalities and ordered elds 4. The complex numbers 5. Alternative de nitions of 6. A glimpse at metric spaces Chapter 2. Complex Numbers 1. Complex conjugation 2. Existence of square roots 3. Limits 4. Convergent in nite series 5. Uniform convergence and consequences 6. The unit circle and trigonometry 7. The geometry of addition and multiplication 8. Logarithms Chapter 3. Complex Numbers and Geometry 1. Lines, circles, and balls 2. Analytic geometry 3. Quadratic polynomials 4. Linear fractional transformations 5. The Riemann sphere Chapter 4. Power Series Expansions 1. Geometric series 2. The radius of convergence 3. Generating functions 4. Fibonacci numbers 5. An application of power series 6. Rationality Chapter 5. Complex Differentiation 1. De nitions of complex analytic function 2. Complex di erentiation 3. The Cauchy-Riemann equations 4. Orthogonal trajectories and harmonic functions 5. A glimpse at harmonic functions 6. What is a di erential form? Chapter 6. Complex Integration 1. Complex-valued functions 2. Line integrals 3. Goursat's proof 4. The Cauchy integral formula 5. A return to the de nition of complex analytic function Chapter 7. Applications of Complex Integration 1. Singularities and residues 2. Evaluating real integrals using complex variables methods 3. Fourier transforms 4. The Gamma function Chapter 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 Further reading Bibliography Index