Math 411: Honours Complex Variables - University of Alberta

Contents
1 The Complex Numbers 5
2 Complex Differentiation 9
3 Power Series 14
4 Complex Line Integrals 22
5 Cauchy’s Integral Theorem and Formula 28
6 Convergence of Holomorphic Functions 42
7 Elementary Properties of Holomorphic Functions 46
8 The Singularities of a Holomorphic Function 52
9 Holomorphic Functions on Annuli 57
10 The Winding Number of a Curve 65
11 The General Cauchy Integral Theorem 68
12 The Residue Theorem and Applications 72
12.1 Applications of the Residue Theorem to Real Integrals . . . . . . . . 75
12.2 The Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . 81
13 Function Theoretic Consequences of the Residue Theorem 87
14 Harmonic Functions 94
15 Analytic Continuation along a Curve 103
16 Montel’s Theorem 107
17 The Riemann Mapping Theorem 110
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