Fundamentals of Real and Complex Analysis (Springer Undergraduate Mathematics Series) 2024th Edition with complete solution

Fundamentals of Real and Complex Analysis (Springer Undergraduate Mathematics Series) 2024th Edition with complete solution Contents Preface vii 1 Introductory Analysis 1 1.1 Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Number Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3 Completeness and the Real Number System . . . . . . . . . . . . 28 1.4 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.5 Topology of the Real Line . . . . . . . . . . . . . . . . . . . . . . 63 1.6 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . 84 1.7 Differentiability on R . . . . . . . . . . . . . . . . . . . . . . . . 98 1.8 The Riemann Integral . . . . . . . . . . . . . . . . . . . . . . . . 111 2 Real Analysis 127 2.1 Metric, Normed, and Inner Product Spaces . . . . . . . . . . . . 127 2.2 Fixed Point Theorems and Applications . . . . . . . . . . . . . . 173 2.3 Modes of Convergence . . . . . . . . . . . . . . . . . . . . . . . . 191 2.4 Approximation by Polynomials . . . . . . . . . . . . . . . . . . . 203 2.5 Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . 213 2.6 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 2.7 Lebesgue Measure and Integration . . . . . . . . . . . . . . . . . 240 2.8 Banach–Tarski Paradox . . . . . . . . . . . . . . . . . . . . . . . 272 3 Complex Analysis 277 3.1 The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . 277 3.2 Holomorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . 289 3.3 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 3.4 Some Holomorphic Functions . . . . . . . . . . . . . . . . . . . . 304 3.5 Conformal Mappings . . . . . . . . . . . . . . . . . . . . . . . . . 315 3.6 Integration in the Complex Plane . . . . . . . . . . . . . . . . . . 331 3.7 Cauchy’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 337 3.8 Cauchy’s Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . 350 3.9 Laurent Expansion and Singularities . . . . . . . . . . . . . . . . 359 3.10 The Bieberbach Conjecture . . . . . . . . . . . . . . . . . . . . . 375 xiiixiv CONTENTS About the Author 383 Bibliography 385