《Complex Analysis》

CHAPTER ⅠCOMPLEX NUMBERS1

1．The algebra of complex numbers1

1．1．Arithmetic operations1

1．2．Square roots2

1．3．Justification4

1．4．Conjugation．Absolute value6

1．5．Inequalities8

2．The geometric representation of complex numbers10

2．1．Geometric addition and multiplication11

2．2．The binomial equation13

2．3．Definition of the argument14

2．4．Straight lines,half planes,and angles18

2．5．The spherical representation20

3．Linear transformations22

3．1．The linear group23

3．2．The cross ratio25

3．3．Symmetry26

3．4．Tangents,orientation,and angles29

3．5．Families of circles31

CHAPTER ⅡCOMPLEX FUNCTIONS36

1．Elementary functions36

1．1．Limits and continuity36

1．2．Analytic functions38

1．3．Rational functions42

1．4．The exponential function46

1．5．The trigonometric functions49

2．Topological concepts51

2．1．Point sets51

2．2．Connected sets56

2．3．Compact sets59

2．4．Continuous functions and mappings61

2．5．Arcs and closed curves64

3．Analytic functions in a region66

3．1．Definition and simple consequences66

3．2．Conformal mapping69

4．Elementary conformal mappings72

4．1．The use of level curves72

4．2．A survey of elementary mappings75

4．3．Elementary Riemann surfaces79

CHAPTER ⅢCOMPLEX INTEGRATION82

1．Fundamental theorems82

1．1．Line integrals82

1．2．Cauchy＇s theorem for a rectangle88

1．3．Cauchy＇s theorem in a circular disk91

2．Cauchy＇s integral formula92

2．1．The index of a point with respect to a closed curve92

2．2．The integral formula95

2．3．Higher derivatives96

3．Local properties of analytic functions99

3．1．Removable singularities．Taylor＇s theorem99

3．2．Zeros and poles102

3．3．The local mapping105

3．4．The maximum principle108

4．The general form of Cauchy＇s theorem111

4．1．Chains and cycles111

4．2．Simple connectivity112

4．3．Exact differentials in simply connected regions114

4．4．Multiply connected regions116

5．The calculus of residues119

5．1．The residue theorem120

5．2．The argument principle123

5．3．Evaluation of definite integrals125

CHAPTER ⅣINFINITE SEQUENCES132

1．Convergent sequences132

1．1．Fundamental sequences132

1．2．Subsequences134

1．3．Uniform convergence135

1．4．Limits of analytic functions137

2．Power series140

2．1．The circle of convergence140

2．2．The Taylor series141

2．3．The Laurent series147

3．Partial fractions and factorization149

3．1．Partial fractions149

3．2．Infinite products153

3．3．Canonical products155

3．4．The gamma function160

3．5．Stirling＇s formula162

4．Normal families168

4．1．Conditions of normality168

4．2．The Riemann mapping theorem172

CHAPTER ⅤTHE DIRICHLET PROBLEM175

1．Harmonic functions175

1．1．Definition and basic properties175

1．2．The mean-value property178

1．3．Poisson＇s formula179

1．4．Harnack＇s principle183

1．5．Jensen＇s formula184

1．6．The symmetry principle189

2．Subharmonic functions193

2．1．Definition and simple properties194

2．2．Solution of Dirichlet＇s problem196

3．Canonical mappings of multiply connected regions199

3．1．Harmonic measures200

3．2．Green＇s function205

3．3．Parallel slit regions206

CHAPTER ⅥMULTIPLE-VALUED FUNCTIONS209

1．Analytic continuation209

1．1．General analytic functions209

1．2．The Riemann surface of a function211

1．3．Analytic continuation along arcs212

1．4．Homotopic curves215

1．5．The monodromy theorem218

1．6．Branch points220

2．Algebraic functions223

2．1．The resultant of two polynomials223

2．2．Definition and properties of algebraic functions224

2．3．Behavior at the critical points226

3．Lineat differential equations229

3．1．Ordinary points230

3．2．Regular singular points232

3．3．Solutions at infinity234

3．4．The hypergeometric differential equation235

3．5．Riemann＇s point of view239

INDEX243