Complex Varables Lecture Notes for Math 122A Summary: Complex Varables Lecture Notes for Math 122A John Douglas Moore July 27, 2011 Contents 1 Complex Numbers 1 1.1 Field axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 The complex exponential . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Subsets of the complex plane . . . . . . . . . . . . . . . . . . . . 11 1.6 The Riemann sphere . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Analytic functions 17 2.1 Convergence and continuity . . . . . . . . . . . . . . . . . . . . . 18 2.2 Complex derivatives and analyticity . . . . . . . . . . . . . . . . 22 2.3 The Cauchy-Riemann equations . . . . . . . . . . . . . . . . . . . 25 2.4 Fluid motion in the plane . . . . . . . . . . . . . . . . . . . . . . 29 3 Examples of analytic functions 35 3.1 Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Linear fractional transformations . . . . . . . . . . . . . . . . . . 39 Collections: Mathematics