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1 Holomorphic functions 1

1.1 Holomorphic functions 1

a.The complex plane 1

b.Functions of a complex variable 5

c.Holomorphic functions 9

1.2 Power series 15

a.Series whose terms are functions 15

b.Power series 16

1.3 Integrals 24

a.Curves 25

b.Integrals 30

c.Cauchy’s integral formula for circles 35

d.Power series expansions 43

1.4 Properties of holomorphic functions 47

a.mth-order derivatives 47

b.Limits of sequences of holomorphic functions 49

c.The Mean Value Theorem and the maximum principle 51

d.Isolated singularities 52

e.Entire functions 58

2 Cauchy’s Theorem 60

2.1 Piecewise smooth curves 60

a.Smooth Jordan curves 60

b.Boundaries of bounded closed regions 65

2.2 Cellular decomposition 74

a.Calls 74

b.Cellular decomposition 80

2.3 Cauchy’s Theorem 101

a.Cauchy’s Theorem 101

b.Cauchy’s integral formula 104

c.Residues 106

d.Evaluation of definite integrals 109

2.4 Differentiability and homology 113

3 Conformal mappings 117

3.1 Conformal mappings 117

3.2 The Riemann sphere 132

a.The Riemann sphere 132

b.Holomorphic functions with an isolated singularity at ∞ 135

c.Local coordinates 137

d.Homogeneous coordinates 138

3.3 Linear fractional transformations 139

a.Linear fractional transformations 139

b.Cross ratio 142

c.Elementary conformal mappings 148

4 Analytic continuation 153

4.1 Analytic continuation 153

a.Analytic continuation 153

b.Analytic continuation by expansion in power series 157

4.2 Analytic continuation along curves 160

4.3 Analytic continuation by integrals 180

4.4 Cauchy’s Theorem （continued） 190

5 Riemann’s Mapping Theorem 200

5.1 Riemann’s Mapping Theorem 200

5.2 Correspondence of boundaries 214

5.3 The principle of reflection 224

a.The principle of reflection 224

b.Modular functions 233

c.Picard’s Theorem 241

d.The Schwarz-Christoffel formula 241

6 Riemann surfaces 247

6.1 Differential forms 247

a.Differential forms 247

b.Line integrals 249

c.Harmonic forms 257

d.Harmonic functions 258

6.2 Riemann surfaces 260

a.Hausdorff spaces 260

b.Definition of Riemann surfaces 263

6.3 Differential forms on a Riemann surface 268

a.Differential forms 268

b.Line integrals 272

c.Locally finite open coverings 275

d.Partition of unity 278

e.Green’s Theorem 284

6.4 Dirichlet’s Principle 294

a.Inner product and norm 294

b.Dirichlet’s Principle 299

c.Analytic functions 314

7 The structure of Riemann surfaces 319

7.1 Planar Riemann surfaces 319

a.Planar Riemann surfaces 319

b.Simply connected Riemann surfaces 329

c.Multiply connected regions 333

7.2 Compact Riemann surfaces 340

a.Cohomology groups 340

b.Structure of compact Riemann surfaces 344

c.Homology groups 360

8 Analytic functions on a closed Riemann surface 376

8.1 Abelian differentials of the first kind 376

a.Harmonic 1-forms of the first kind 376

b.Abelian differential of the first kind 379

8.2 Abelian differentials of the second and third kind 379

a.Meromorphic functions 379

b.Abelian differentials of the second and third kind 380

8.3 The Riemann-Roch Theorem 381

a.Existence Theorem 381

b.The Riemann-Roch Theorem 382

8.4 Abel’s Theorem 389

a.Existence Theorem 389

b.Abel’s Theorem 391

Problems 394

Index 401