Appendix A: List of Theorems

A List of Theorems

Definition 1.1.1 (Open Ball)4

Definition 1.1.2 (Open and Closed Sets)4

Definition 1.1.3 (Accumulation Point)4

Theorem 1.1.44

Definition 1.1.5 (Closure)5

Definition 1.1.6 (Interior)5

Definition 1.1.7 (Set Covering)5

Definition 1.1.8 (Compact Set)5

Theorem 1.1.9 (Heine–Borel)5

Theorem 1.1.10 (Bolzano–Weierstrass)6

Definition 1.1.11 (Support of a Function)7

Theorem 1.2.1 (Fundamental Theorem of Calculus, Differential Form)8

Theorem 1.2.2 (Fundamental Theorem of Calculus, Integral Form)9

Theorem 1.2.3 (Mean Value Theorem, Differential Form)9

Theorem 1.2.4 (Mean Value Theorem, Integral Form)9

Theorem 1.2.5 (Jordan Curve Theorem)10

Definition 1.2.610

Example 1.2.711

Theorem 1.2.8 (Leibniz Integral Rule)11

Theorem 1.2.9 (Gradient Theorem)11

Theorem 1.2.10 (Green's Theorem)11

Theorem 1.2.11 (Stokes’ Theorem)11

Theorem 1.2.12 (Gauss’ Theorem)12

Lemma 1.2.13 (Poincaré)13

Theorem 1.2.14 (Stokes–Cartan)13

Definition 1.2.15 (Epsilon–Delta)14

Definition 1.2.16 (Epsilon–N)14

Theorem 1.2.17 (Cauchy Criterion)14

Definition 1.2.18 (Limit Superior)14

Definition 1.2.19 (Limit Inferior)15

Lemma 1.2.2015

Definition 1.2.21 (Continuity)15

Theorem 1.2.2216

Theorem 1.2.23 (Extreme Value)16

Definition 1.2.24 (Uniform Continuity)16

Example 1.2.2516

Theorem 1.2.26 (Heine–Cantor)17

Definition 1.2.2718

Proposition 1.2.2818

Proposition 1.2.2918

Definition 2.2.1 (Holomorphy)22

Definition 2.2.222

Theorem 2.2.322

Definition 2.2.4 (Harmonicity)23

Proposition 2.2.524

Theorem 2.2.1.1 (Chain Rule)25

Theorem 2.2.1.227

Definition 2.3.1 (Uniform Convergence)28

Theorem 2.3.2 (Cauchy Criterion)28

Theorem 2.3.3 (Weierstrass $𝑀$ -Test)28

Theorem 2.3.4 (Abel’s Theorem)29

Theorem 2.3.5 (Cauchy–Hadamard)30

Theorem 2.3.6 (Uniform Limit)31

Theorem 2.3.732

Theorem 2.5.1 (De Moivre)35

Theorem 3.1.1 (Lusin Area Theorem)41

Theorem 3.1.2 (Green’s Theorem, Complex Form)41

Theorem 3.1.3 (Cauchy–Pompeiu)42

Corollary 3.1.3.144

Theorem 3.1.4 (Cauchy’s Integral Formula)45

Theorem 3.1.5 (Cauchy’s Integral Theorem)45

Theorem 3.1.6 (Solution to the one-dimensional $\overline{𝜕}$ -problem)46

Lemma 3.1.748

Lemma 3.1.8 (Goursat)49

Theorem 3.1.9 (Cauchy–Goursat)51

Theorem 3.1.10 (Cauchy–Goursat)54

Example 3.1.1155

Example 3.1.1256

Proposition 3.1.13 (Cauchy–Goursat Formula on the Exterior)57

Theorem 3.2.1 (Cauchy–Goursat)58

Example 3.2.261

Theorem 3.2.3 (Cauchy’s Estimate)61

Example 3.2.462

Theorem 3.2.5 (Liouville)63

Theorem 3.2.6 (Morera)64

Theorem 3.2.764

Corollary 3.2.7.166

Definition 3.2.8 (Analytic Continuation)66

Theorem 3.2.9 (Riemann)67

Definition 3.2.1.1 (Topological Space)68

Definition 3.2.1.268

Definition 3.2.1.368

Proposition 3.2.1.468

Proposition 3.2.1.569

Definition 3.2.1.669

Proposition 3.2.1.769

Definition 3.2.1.870

Example 3.2.1.970

Proposition 3.2.1.1070

Definition 3.2.1.11 (Basis for a Topology)71

Definition 3.2.1.12 (Metric Space)71

Definition 3.2.1.1372

Proposition 3.2.1.1472

Theorem 3.2.1.1572

Theorem 3.2.1.16 (Compact–Hausdorff)73

Example 3.2.1.1773

Definition 3.2.1.1874

Theorem 3.2.1.19 (Connectivity Argument)75

Example 3.2.1.2075

Example 3.2.1.2175

Proposition 3.2.1.2275

Definition 3.2.1.23 (Path Connectivity)76

Theorem 3.2.1.2476

Example 3.2.1.2576

Definition 3.2.1.26 (Exhaustion by Compact Sets)77

Lemma 3.2.1.2777

Theorem 3.2.1.28 (Partition of Unity)79

Theorem 3.2.1.29 (Existence of Bump Functions)81

Theorem 3.3.1 (Fundamental Theorem of Algebra)82

Theorem 3.3.282

Theorem 3.3.3 (Identity Theorem)84

Theorem 3.3.4 (Holomorphic Argument Principle)84

Theorem 3.3.586

Theorem 3.3.6 (Rouché)88

Theorem 3.3.789

Theorem 3.3.8 (Hurwitz)89

Lemma 3.4.190

Theorem 3.4.2 (Maximum Modulus Principle)90

Theorem 3.4.392

Theorem 3.4.492

Theorem 3.4.5 (Maximum Principle for Harmonic Functions)93

Lemma 3.5.1 (Schwarz)93

Definition 3.5.2 (Group)94

Theorem 3.5.3 (The Holomorphic Automorphism Group on $𝔻$ )95

Lemma 3.5.4 (Schwarz–Pick)96

Theorem 3.6.1.1103

Theorem 3.6.2.1108

Theorem 4.1.1 (Weierstrass)109

Proposition 4.1.2110

Theorem 4.1.3111

Theorem 4.2.1114

Example 4.2.2115

Theorem 4.2.3115

Theorem 4.2.4 (Casorati–Sokhotski–Weierstrass)116

Theorem 4.2.5116

Example 4.2.1.1118

Proposition 4.3.1118

Example 4.3.2119

Definition 4.3.3 (Meromorphy)119

Theorem 4.3.4120

Theorem 4.4.1122

Theorem 4.4.2 (Argument Principle)123

Theorem 4.4.1.1 (The Holomorphic Automorphism Group on $ℂ$ )124

Definition 4.4.2.1 (Coset)125

Theorem 4.4.2.2125

Definition 4.4.2.3 (Group Homomorphism)126

Definition 4.4.2.4 (Group Isomorphism)126

Theorem 4.4.2.5 (The Meromorphic Automorphism Group on $\hat{ℂ}$ )128

Proposition 4.4.2.6128

Lemma 4.4.3.1129

Lemma 4.4.3.2130

Lemma 4.4.3.3130

Theorem 4.4.3.4 (Weierstrass Product Theorem)135

Theorem 4.4.3.5 (Weierstrass Factorization Theorem)135

Corollary 4.4.3.5.1136

Theorem 4.4.3.6 (Mittag–Leffler)137

Theorem 4.4.3.7140

Theorem 4.4.3.8140

Example 4.4.3.9142

Lemma 4.4.4.1143

Theorem 4.4.4.2 (Jensen’s Formula)143

Corollary 4.4.4.2.1 (Jensen’s Inequality)145

Theorem 4.4.4.3 (Poisson–Jensen Formula)145

Lemma 4.4.4.4146

Theorem 4.4.4.5 (Blaschke Product)147

Corollary 4.4.4.5.1148

Definition 4.4.4.6 (Growth Order of Entire Functions)149

Proposition 4.4.4.7150

Theorem 4.4.4.8150

Example 4.4.4.9151

Lemma 4.4.4.10152

Theorem 4.4.4.11152

Theorem 4.4.4.12153

Definition 4.4.4.13154

Definition 4.4.4.14 (Weierstrass Canonical Factorization)154

Lemma 4.4.4.15155

Theorem 4.4.4.16156

Proposition 4.4.4.17157

Lemma 4.4.4.18 (Logarithmic Factorization)158

Theorem 4.4.5.1159

Corollary 4.4.5.1.1159

Proposition 4.4.5.2160

Theorem 4.4.5.3 (Hadamard Factorization Theorem)161

Theorem 4.4.5.4162

Definition 4.5.1 (Residue)163

Theorem 4.5.2 (Residue Theorem)164

Theorem 4.5.3 (Global Residue Theorem)164

Example 4.5.4166

Example 4.5.5 (Dirichlet Integral)167

Example 4.5.6 (Fresnel Integrals)169

Example 4.5.7171

Example 4.5.8171

Example 4.5.9173

Theorem 4.5.10176

Example 4.5.11177

Theorem 5.1.1 (Open Mapping Theorem)179

Lemma 5.1.2179

Theorem 5.1.3180

Theorem 5.1.4180

Theorem 5.1.5181

Example 5.1.6182

Example 5.1.7182

Theorem 5.1.8183

Definition 5.1.9 (Cross-Ratio)184

Definition 5.2.1 (Normal Family)185

Definition 5.2.2 (Equicontinuity)185

Theorem 5.2.3186

Theorem 5.2.4 (Arzelà–Ascoli)186

Definition 5.2.5189

Theorem 5.2.6 (Montel’s Theorem)189

Definition 5.3.1 (Holomorphic Logarithms)191

Theorem 5.3.2 (Riemann Mapping Theorem)191

Example 5.4.1200

Theorem 5.5.1 (Painlevé)204

Theorem 5.5.2 (Schwarz Reflection Principle)205

Theorem 5.5.3 (Symmetry Principle)206

Proposition 6.1.1207

Lemma 6.1.2 (Pole-Pushing Lemma)209

Lemma 6.1.3 (Generalized Pole-Pushing Lemma)212

Theorem 6.1.4 (Runge)213

Definition 6.2.1215

Proposition 6.2.2216

Proposition 6.2.3216

Proposition 6.2.4219

Theorem 6.2.5 (Tietze–Urysohn–Brouwer)221

Corollary 6.2.5.1222

Proposition 6.2.6223

Theorem 6.2.7 (Mergelyan)225

Definition 6.3.1 (Analytic Capacity)230

Proposition 6.3.2230

Theorem 6.3.3230

Proposition 6.3.4231

Proposition 6.3.5231

Proposition 6.3.6232

Theorem 6.3.7232

Corollary 6.3.7.1232

Example 6.3.8232

Theorem 7.1.1235

Theorem 7.1.2 (Elliptic Regularity)235

Theorem 7.1.3 (Minimum Principle for Harmonic Functions)236

Theorem 7.1.4 (Schwarz Reflection Principle for Harmonic Functions)236

Theorem 7.2.1238

Theorem 7.2.2 (Harnack Inequality)238

Theorem 7.2.3 (Harnack’s Principle)239

Example 7.3.1241

Definition 7.3.2 (Subharmonicity)242

Proposition 7.3.3242

Proposition 7.3.4243

Theorem 7.3.5243

Corollary 7.3.5.1246

Theorem 7.3.6246

Theorem 7.3.7 (Jensen’s Inequality)248

Theorem 7.3.8249

Theorem 7.3.9250

Theorem 7.3.10 (Maximum Principle for Subharmonic Functions)250

Proposition 7.3.11252

Definition 7.3.12 (Barrier)252

Theorem 7.3.13252

Theorem 7.3.14253

Theorem 7.3.15253

Theorem 7.3.16254

Proposition 7.3.17255

Proposition 7.4.1 (Boundary Maximum Principle)257

Theorem 7.4.2 (Solution to the Dirichlet Problem)258

Definition 7.5.1 (Conformal Modulus of an Annulus)263

Theorem 7.5.2263

Corollary 7.5.2.1 (Holomorphic Automorphism Group of an Annulus)266

Lemma 7.5.3 (Polygonal Grid Exhaustion)267

Theorem 7.5.4276

Theorem 8.3.1 (Schwarz–Ahlfors–Pick)296

Theorem 8.3.2297

Theorem 8.3.3298

Corollary 8.3.3.1299

Corollary 8.3.3.2 (Generalized Liouville)300

Proposition 8.3.4300

Proposition 8.3.5301

Theorem 8.3.6 (Picard’s Little Theorem)302

Definition 8.4.1303

Definition 8.4.2303

Definition 8.4.3 (Spherical Derivative)303

Proposition 8.4.4304

Proposition 8.4.5304

Theorem 8.4.6 (Marty’s Criterion)306

Theorem 8.4.7 (Fundamental Normality Test)309

Corollary 8.4.7.1 (Montel–Carathéodory)310

Theorem 8.5.1 (Picard’s Great Theorem)311

Corollary 8.5.1.1313

Theorem 8.5.2 (Bloch)314

Theorem 8.5.3 (Landau)314

Theorem 8.5.4314

Theorem 8.5.5 (Landau–Carathéodory)316

Theorem 8.5.6 (Schottky)316

Theorem 9.1.1 (Liouville)319

Proposition 9.1.2320

Proposition 9.1.3320

Theorem 9.1.4320

Proposition 9.1.5321

Definition 9.1.6321

Proposition 9.1.7321

Theorem 9.1.8321

Proposition 9.1.1.1322

Proposition 9.1.1.2323

Definition 9.1.1.3 (Weierstrass $℘$ -Function)324

Proposition 9.1.1.4325

Definition 9.1.1.5 (Weierstrass $𝜁$ -Function)326

Definition 9.1.1.6 (Weierstrass $𝜎$ -Function)326

Definition 9.2.1326

Lemma 9.2.2327

Theorem 9.2.3 (Area Principle)329

Theorem 9.2.4331

Theorem 9.2.5332

Theorem 9.2.6 (Koebe Quarter Theorem)332

Example 9.3.1333

Example 9.3.2335

Theorem 9.3.3 (Osgood–Taylor–Carathéodory)335

Lemma 9.3.4336

Proposition 9.3.5337

Lemma 9.3.6338

Lemma 9.3.7339

Proposition 9.3.8340

Lemma 9.3.9340

Lemma 9.3.10341

Corollary 9.3.10.1342

Proposition 9.3.11342

Lemma 9.3.12343

Proposition 9.3.13344

Theorem 9.3.14344

Definition 9.4.1 (Hardy Space)348

Theorem 9.4.2348

Proposition 9.4.3348

Lemma 9.4.4 (Hardy)348

Theorem 9.4.5349

Theorem 9.4.6351

Theorem 9.4.7352

Definition 10.1.1 (Inner Product)355

Proposition 10.1.2355

Example 10.1.3356

Example 10.1.4356

Definition 10.1.5 (Associated Norm)356

Definition 10.1.6 (Isometry)357

Definition 10.1.7 (Isomorphism)357

Example 10.1.8357

Theorem 10.1.9 (Cauchy–Schwarz Inequality)357

Theorem 10.1.10 (Triangle Inequality of the Associated Norm)358

Theorem 10.1.11 (Continuity of the Inner Product)358

Proposition 10.1.12359

Definition 10.1.13 (Hilbert Space)360

Example 10.1.14360

Example 10.1.15 (Bergman Space)360

Proposition 10.1.16361

Example 10.1.17363

Theorem 10.1.18 (Parallelogram Law)363

Definition 10.1.19 (Convexity)364

Definition 10.1.20 (Orthogonality)364

Definition 10.1.21364

Theorem 10.1.22364

Definition 10.1.23 (Orthogonal Complement)365

Definition 10.1.24 (Vector Projection)365

Proposition 10.1.25365

Proposition 10.1.26365

Proposition 10.1.27366

Theorem 10.1.28 (Orthogonal Decomposition)366

Corollary 10.1.28.1367

Definition 10.1.29 (Linear Functional)367

Definition 10.1.30367

Definition 10.1.31368

Theorem 10.1.32368

Definition 10.1.33 (Operator Norm)368

Proposition 10.1.34368

Definition 10.1.35 (Kernel)369

Theorem 10.1.36 (Riesz Representation Theorem)369

Corollary 10.1.36.1370

Definition 10.1.1.1 (Basis and Dimension)370

Proposition 10.1.1.2370

Definition 10.1.1.3371

Definition 10.1.1.4 (Orthonormal Basis)371

Theorem 10.1.1.5 (Gram–Schmidt Orthonormalization)372

Corollary 10.1.1.5.1373

Definition 10.2.1 (Orthonormal System)373

Definition 10.2.2 (Completeness of an Orthonormal System)374

Theorem 10.2.3374

Corollary 10.2.3.1 (Bessel’s Inequality)375

Theorem 10.2.4 (Riesz–Fischer)376

Corollary 10.2.4.1378

Example 10.2.5379

Example 10.2.6380

Definition 10.2.7 (Separability)382

Proposition 10.2.8382

Theorem 10.2.9383

Definition 10.3.1 (Bergman Kernel)384

Theorem 10.3.2 (Conjugate-symmetry of the Bergman Kernel)385

Corollary 10.3.2.1385

Theorem 10.3.3 (Reproducing Property of $𝐾 (\cdot, \cdot)$ )385

Theorem 10.3.4386

Definition 11.1.1388

Theorem 11.1.2 (Cauchy’s Integral Formula on Polydisks)388

Corollary 11.1.2.1 (Cauchy’s Estimate on Polydisks)389

Theorem 11.1.3390

Theorem 11.1.4 (Identity)391

Theorem 11.1.5 (Maximum Modulus Principle)391

Theorem 11.1.6 (Weierstrass)392

Theorem 11.1.7 (Montel)392

Definition 11.2.1 (Multi-Index Notation)392

Definition 11.2.2392

Proposition 11.2.3393

Lemma 11.2.4 (Cartan)393

Definition 11.2.5 (Reinhardt Domain)395

Definition 11.2.6396

Definition 11.2.7 (Circular Domain)396

Definition 11.2.8396

Proposition 11.2.9396

Lemma 11.2.10 (Cartan)397

Theorem 11.2.11 (The Holomorphic Automorphism Group on $𝔻^{𝑛}$ )398

Definition 11.2.12400

Definition 11.2.13400

Definition 11.2.14400

Theorem 11.2.15 (Spectral Theorem)400

Proposition 11.2.16401

Proposition 11.2.17402

Definition 11.2.18403

Proposition 11.2.19403

Theorem 11.2.20 (The Holomorphic Automorphism Group on $𝐵^{𝑛}$ )404

Lemma 11.3.1404

Theorem 11.3.2 (Poincaré)405

Theorem 11.4.1 (Hartogs’ Extension Theorem)408

Proposition 11.4.2408

Proposition 11.4.3410

Theorem 11.4.4 (Hartogs’ Extension Theorem for Reinhardt Domains)410

Example 11.4.5411

Definition 12.1.1412

Theorem 12.1.2412

Theorem 12.1.3 (Gauss)413

Theorem 12.1.4 (Weierstrass)415

Theorem 12.1.5 (Euler’s Reflection Formula)416

Example 12.1.6416

Theorem 12.1.7 (Legendre’s Duplication Formula)417

Theorem 12.1.8 (Gauss Multiplication Theorem)417

Definition 12.1.9 (Beta Function)418

Theorem 12.1.10418

Example 12.1.11 (MIT Integration Bee 2023 Finals #1)418

Example 12.1.12419

Definition 12.2.1420

Theorem 12.2.2 (Euler Product Formula)421

Theorem 12.2.3421

Theorem 12.2.4425

Theorem 12.2.5 (Riemann’s Reflection Formula)426

Proposition 12.2.6429

Conjecture 12.2.7 (Riemann Hypothesis)429

Definition 12.2.8430

Proposition 12.2.9430

Proposition 12.2.10430

Theorem 12.2.11430

Theorem 12.3.1433

Theorem 12.3.2434

Lemma 12.3.1.1436

Definition 12.3.1.2438

Proposition 12.3.1.3438

Lemma 12.3.1.4 (Riemann–Lebesgue)440

Theorem 12.3.1.5 (Wiener–Ikehara)442

Theorem 12.3.1.6 (Prime Number Theorem)445

Definition 13.1.1446

Theorem 13.1.2446

Definition 13.1.3447

Example 13.1.4447

Example 13.1.5448

Example 13.1.6448

Example 13.1.7448

Definition 13.2.1449

Lemma 13.2.2449

Definition 13.3.1451

Definition 13.3.2451

Theorem 13.3.3451

Corollary 13.3.3.1453

Definition 14.1.1455

Proposition 14.1.2 (Properties of ${log}^{+}$ )456

Definition 14.1.3456

Definition 14.1.4457

Proposition 14.1.5457

Proposition 14.2.1457

Theorem 14.2.2458

Theorem 14.2.3459

Theorem 14.2.4 (First Fundamental Theorem of Nevanlinna Theory)460

Corollary 14.2.4.1461

Proposition 14.2.5461

Theorem 14.2.1.1463

Definition 14.2.1.2464

Theorem 14.2.1.3465

Proposition 14.2.1.4465

Theorem 14.2.1.5466

Theorem 14.2.2.1467

Lemma 14.3.1468

Definition 14.3.2 (Ahlfors–Shimizu Proximity Function)472

Definition 14.3.3 (Ahlfors–Shimizu Characteristic Function)472

Theorem 14.3.4472

Theorem 14.3.5 (First Fundamental Theorem in Ahlfors–Shimizu Form)474

Theorem 14.3.6476

Theorem 14.4.1 (Second Fundamental Theorem of Nevanlinna Theory, Part 1)480

Lemma 14.4.1.1484

Lemma 14.4.1.2485

Proposition 14.4.1.3486

Lemma 14.4.1.4 (Borel)490

Theorem 14.4.1.5 (Second Fundamental Theorem of Nevanlinna Theory, Part 2)492

Proposition 14.4.1.6495

Definition 14.4.1.7495

Theorem 14.4.1.8496

Definition 14.5.1.1497

Definition 14.5.1.2497

Theorem 14.5.1.3 (Deficiency Relation)498

Corollary 14.5.1.3.1500

Corollary 14.5.1.3.2 (Picard’s Little Theorem)500

Theorem 14.5.2.1 (Nevanlinna’s Five-Value Theorem)500