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CHAPTER Ⅰ COMPLEX NUMBERS 1

1．The algebra of complex numbers 1

1．1．Arithmetic operations 1

1．2．Square roots 2

1．3．Justification 4

1．4．Conjugation．Absolute value 6

1．5．Inequalities 8

2．The geometric representation of complex numbers 10

2．1．Geometric addition and multiplication 11

2．2．The binomial equation 13

2．3．Definition of the argument 14

2．4．Straight lines,half planes,and angles 18

2．5．The spherical representation 20

3．Linear transformations 22

3．1．The linear group 23

3．2．The cross ratio 25

3．3．Symmetry 26

3．4．Tangents,orientation,and angles 29

3．5．Families of circles 31

CHAPTER Ⅱ COMPLEX FUNCTIONS 36

1．Elementary functions 36

1．1．Limits and continuity 36

1．2．Analytic functions 38

1．3．Rational functions 42

1．4．The exponential function 46

1．5．The trigonometric functions 49

2．Topological concepts 51

2．1．Point sets 51

2．2．Connected sets 56

2．3．Compact sets 59

2．4．Continuous functions and mappings 61

2．5．Arcs and closed curves 64

3．Analytic functions in a region 66

3．1．Definition and simple consequences 66

3．2．Conformal mapping 69

4．Elementary conformal mappings 72

4．1．The use of level curves 72

4．2．A survey of elementary mappings 75

4．3．Elementary Riemann surfaces 79

CHAPTER Ⅲ COMPLEX INTEGRATION 82

1．Fundamental theorems 82

1．1．Line integrals 82

1．2．Cauchy＇s theorem for a rectangle 88

1．3．Cauchy＇s theorem in a circular disk 91

2．Cauchy＇s integral formula 92

2．1．The index of a point with respect to a closed curve 92

2．2．The integral formula 95

2．3．Higher derivatives 96

3．Local properties of analytic functions 99

3．1．Removable singularities．Taylor＇s theorem 99

3．2．Zeros and poles 102

3．3．The local mapping 105

3．4．The maximum principle 108

4．The general form of Cauchy＇s theorem 111

4．1．Chains and cycles 111

4．2．Simple connectivity 112

4．3．Exact differentials in simply connected regions 114

4．4．Multiply connected regions 116

5．The calculus of residues 119

5．1．The residue theorem 120

5．2．The argument principle 123

5．3．Evaluation of definite integrals 125

CHAPTER Ⅳ INFINITE SEQUENCES 132

1．Convergent sequences 132

1．1．Fundamental sequences 132

1．2．Subsequences 134

1．3．Uniform convergence 135

1．4．Limits of analytic functions 137

2．Power series 140

2．1．The circle of convergence 140

2．2．The Taylor series 141

2．3．The Laurent series 147

3．Partial fractions and factorization 149

3．1．Partial fractions 149

3．2．Infinite products 153

3．3．Canonical products 155

3．4．The gamma function 160

3．5．Stirling＇s formula 162

4．Normal families 168

4．1．Conditions of normality 168

4．2．The Riemann mapping theorem 172

CHAPTER Ⅴ THE DIRICHLET PROBLEM 175

1．Harmonic functions 175

1．1．Definition and basic properties 175

1．2．The mean-value property 178

1．3．Poisson＇s formula 179

1．4．Harnack＇s principle 183

1．5．Jensen＇s formula 184

1．6．The symmetry principle 189

2．Subharmonic functions 193

2．1．Definition and simple properties 194

2．2．Solution of Dirichlet＇s problem 196

3．Canonical mappings of multiply connected regions 199

3．1．Harmonic measures 200

3．2．Green＇s function 205

3．3．Parallel slit regions 206

CHAPTER Ⅵ MULTIPLE-VALUED FUNCTIONS 209

1．Analytic continuation 209

1．1．General analytic functions 209

1．2．The Riemann surface of a function 211

1．3．Analytic continuation along arcs 212

1．4．Homotopic curves 215

1．5．The monodromy theorem 218

1．6．Branch points 220

2．Algebraic functions 223

2．1．The resultant of two polynomials 223

2．2．Definition and properties of algebraic functions 224

2．3．Behavior at the critical points 226

3．Lineat differential equations 229

3．1．Ordinary points 230

3．2．Regular singular points 232

3．3．Solutions at infinity 234

3．4．The hypergeometric differential equation 235

3．5．Riemann＇s point of view 239

INDEX 243