Course 214 Section 1: Basic Theorems of Complex Analysis Second Semester 2008

1 Basic Theorems of Complex Analysis 1 1.1 The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Infinite Sequences of Complex Numbers . . . . . . . . . . . . . 3 1.3 The Least Upper Bound Principle . . . . . . . . . . . . . . . . 5 1.4 Monotonic Sequences . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Upper and Lower Limits of Bounded Sequences of Real Numbers 7 1.6 The Bolzano-Weierstrass Theorem . . . . . . . . . . . . . . . . 8 1.7 Cauchy’s Criterion for Convergence . . . . . . . . . . . . . . . 9 1.8 Limits of Functions of a Complex Variable . . . . . . . . . . . 10 1.9 Continuous Functions of a Complex Variable. . . . . . . . . . 12 1.10 The Intermediate Value Theorem . . . . . . . . . . . . . . . . 14 1.11 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . 15 1.12 Open Sets in the Complex Plane . . . . . . . . . . . . . . . . 16 1.13 Interiors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.14 Closed Sets in the Complex Plane . . . . . . . . . . . . . . . . 19 1.15 Closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.16 Continuous Functions and Open and Closed Sets . . . . . . . 21 1.17 Continuous Functions on Closed Bounded Sets . . . . . . . . . 22 1.18 Uniform Continuity . . . . . . . . . . . . . . . . . . . . . . . . 23