An Introduction to Complex Analysis

Preface
vii
Chapter
1
Basic properties of sets and functions in the com-
plex plane
1
§1
Metric properties of the complex plane
1
§2
Differentiation and integration of complex func-
tions
10
Exercises
24
Chapter 2
Cauchy’s theorem
27
§1
Cauchy’s theorem for a starred domain
27
§2
Integral formulae and higher derivatives
31
§3
Morera’s and Liouville’s theorems
33
Exercises
36
Chapter 3
Local properties of regular functions
39
§1
Taylor’s theorem
39
§2
Laurent expansions
45
Exercises
49
Chapter 4
Zeros and singularities of regular functions
53
§1
Classification of zeros and isolated singularities
53
§2
Residues
59
Exercises
61
Chapter 5
The residue theorem
63
§1
The topological index
63
§2
The
residue theorem
69
§3
Rouche’s theorem and the local mapping theorem
83
Exercises
90
Chapter 6
Harmonic functions and the Dirichlet problem
94
§1
Harmonic functions
94
§2
Harmonic conjugates
100
Exercises
105
Appendix A:
The regulated integral
107
Appendix B:
Some topological considerations
119
§1
Simple connectedness
119
§2
The Jordan curve theorem
127
Appendix C:
Logarithms and fractional powers
129
Bibliography
137
Index
139
Index of Notations
141