Notes on Complex Analysis

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Chapter 1: Prerequisites
§1.1 Topological Preliminaries
§1.2 Calculus
Chapter 2: Introduction to Complex Functions
§2.1 The Extended Complex Plane and its Spherical Representation
§2.2 Complex Differentiation
§2.2.1 Wirtinger Derivatives
§2.3 Complex Power Series
§2.4 The Conformality of Holomorphic Mappings
§2.5 Elementary Functions
Chapter 3: Complex Integration
§3.1 The Cauchy–Goursat Theorem
§3.2 Analyticity and Holomorphy
§3.2.1 Topology, Partitions of Unity, and the Existence of Bump Functions
§3.3 Zeros of a Holomorphic Function
§3.4 Further Properties of Holomorphic Functions
§3.5 The Group of Holomorphic Automorphisms on the Unit Disk
§3.6 Alternative Integral Formulas
§3.6.1 Solution to the Dirichlet Problem on a Disk
§3.6.2 The Poisson Integral Formula in Harmonic Analysis
Chapter 4: The Theory of Weierstrass
§4.1 Laurent Series
§4.2 Isolated Singularities
§4.2.1 Isolated Singularities at the $\infty$ Point
§4.3 Entireness and Meromorphy
§4.4 Further Properties of Meromorphic and Entire Functions
§4.4.1 The Group of Holomorphic Automorphisms on $ℂ$
§4.4.2 The Group of Meromorphic Automorphisms on $\hat{ℂ}$
§4.4.3 The Construction of Entire and Meromorphic Functions
§4.4.4 Classifying Growth of Entire Functions
§4.4.5 Hadamard Factorization Theorem
§4.5 The Residue Theorem
Chapter 5: The Geometric Theory of Conformal Mappings
§5.1 Biholomorphy
§5.2 Normal Families
§5.3 The Riemann Mapping Theorem
§5.4 The Schwarz–Christoffel Transformation
§5.5 The Reflection Principle
Chapter 6: Rational Approximation Theory
§6.1 Runge’s Theorem
§6.2 Mergelyan’s Theorem
§6.3 Analytic Capacity
Chapter 7: Harmonic Functions
§7.1 Basic Properties of Harmonic Functions
§7.2 Harnack’s Principle
§7.3 Subharmonic Functions
§7.4 The Perron Method and the Solution to the Dirichlet Problem
§7.5 Biholomorphisms between Annuli and the General Riemann Mapping Theorem
Chapter 8: Differential Geometry
§8.1 Gaussian Curvature of a Surface
§8.2 Conformal Metrics and Curvature
§8.3 From Schwarz–Pick to Ahlfors and Value Distribution of Entire Functions
§8.4 A Spherical Generalization of Normal Families
§8.5 The Great Picard, Bloch, Landau, and Schottky Theorems
Chapter 9: Special Classes of Holomorphic Functions
§9.1 Elliptic Functions
§9.1.1 Weierstrass Elliptic Functions
§9.1.2 Jacobi Elliptic Functions
§9.1.3 The Modular Group
§9.2 Schlicht Functions
§9.3 Boundary Continuity of Biholomorphisms
§9.4 Hardy Spaces
Chapter 10: Hilbert Spaces of Holomorphic Functions
§10.1 Geometry of a Hilbert Space
§10.1.1 The Gram–Schmidt Process
§10.2 Orthonormal Systems of a Hilbert Space
§10.3 The Bergman Kernel
Chapter 11: A Glimpse into the Function Theory of Multiple Complex Variables
§11.1 Consequences of Holomorphy
§11.2 The Group of Holomorphic Automorphisms on $𝔻^{𝑛}$ and $𝐵^{𝑛}$
§11.3 Topological Equivalence and Biholomorphic Equivalence
§11.4 Hartogs’ Phenomenon
Chapter 12: Precise Connections to Number Theory
§12.1 The $Γ$ -Function
§12.2 The Riemann $𝜁$ -Function
§12.3 Prime Number Theorem
§12.3.1 The Wiener–Ikehara Theorem
Chapter 13: Analytic Continuation and Riemann Surfaces
§13.1 Analytic Function Elements
§13.2 Analytic Continuation Along a Curve
§13.3 The Monodromy Theorem
Chapter 14: An Introduction to Nevanlinna Theory
§14.1 Nevanlinna’s Counting, Proximity, and Characteristic Functions
§14.2 Properties of $𝑇 (𝑟, 𝑓)$ and the First Fundamental Theorem
§14.2.1 Classifying Growth of Meromorphic Functions
§14.2.2 The Factorization of Transcendental Meromorphic Functions
§14.3 The Ahlfors–Shimizu Characteristic
§14.4 The Second Fundamental Theorem
§14.4.1 The Estimation of $𝑆 (𝑟, 𝑓)$
§14.5 Corollaries to the Second Fundamental Theorem
§14.5.1 Deficiency Relation
§14.5.2 Functions Sharing Values
Appendix A: List of Theorems
Appendix B: Table of Figures
Bibliography

§14.5 Corollaries to the Second Fundamental Theorem

14.5 Corollaries to the Second Fundamental Theorem

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𝑆 (𝑟, 𝑓)

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