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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

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics