Preface

Chapter 1: Analytic Continuation

1. The Exponential Function and the Logarithm

2. Continuation Sequences

3. Continuation Along an Arc

4. Germs

5. Existence of Continuations

6. The Winding Number

7. The Argument Principle

8. The Monodromy Theorem

9. Composition of Germs

10. Composition of Continuations

11. Covering Surfaces

Chapter 2 Geometric Considerations

1. Complex Projective Space

2. Linear Transformations

3. Fractional Linear Transformations

4. Properties of Fractional Linear Transformations

5. Symmetry

6. Schwarz's Lemma

7. Non-Euclidean Geometry

8. The Schwarz Reflection Principle

Chapter 3 The Mapping Theorems of Riemann And Koebe

1. Analytic Equivalence

2. Local Uniform Convergence

3. A Theorem of Hurwitz

4. Implications of Pointwise Convergence

5. Implications of Convergence on a Subset

6. Approximately Linear Functions — Another Application of Schwarz's Lemma

7. A Uniformization Theorem

8. A Closer Look At the Covering

9. Boundary Behavior

10. Lindelöf's Lemma

11. Facts From Topology

12. Continuity at the Boundary

13. A Theorem of Fejér

Chapter 4 The Modular Function

1. Exceptional Values

2. The Modular Configuration

3. The Landau Radius

4. Schottky's Theorem

5. Normal Families

6. Montel's Theorem

7. Picard's Second Theorem

8. The Koebe-Faber Distortion Theorem

9. Bloch's Theorem

Chapter 5 The Hadamard Product Theorem

1. Infinite Products

2. Products of Functions

3. The Weierstrass Product Theorem

4. Functions of Finite Order

5. Exponent of Convergence

6. Canonical Products

7. The Borel-Carathéodory Lemma — Another Form of Schwarz's Lemma

8. A Lemma of H. Cartan

9. The Hadamard Product Theorem

10. The Gamma Function

11. Standard Formulas

12. The Integral Representation of Γ(z)

Chapter 6 The Prime Number Theorem

1. Dirichlet Series

2. Number-Theoretic Functions

3. Statement of the Prime Number Theorem

4. The Riemann Zeta Function

5. Analytic Continuation of ζ(s)

6. Riemann's Functional Equation

7. The Zeros Of ζ(s) In The Critical Strip

8. ζ(s) for Re(s) = 1

9. Integral Representation of Dirichlet Series

10. Integral-Theoretic Lemmas

11. Weak Limits

12. A Tauberian Theorem

Bibliography

Index