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Main Question or Discussion Point
- Author: Serge Lang
- Title: Complex Analysis
- Amazon Link: https://www.amazon.com/dp/0387985921/?tag=pfamazon01-20
- Prerequisities: Basic analysis
- Level: Grad
Table of Contents:
Code:
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[*] Foreword
[*] Prerequisites
[*] Basic Theory
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[*] Complex Numbers and Functions
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[*] Definition
[*] Polar Form
[*] Complex Valued Functions
[*] Limits and Compact Sets
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[*] Compact Sets
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[*] Complex Differentiability
[*] The Cauchy-Riemann Equations
[*] Angles Under Holomorphic Maps
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[*] Power Series
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[*] Formal Power Series
[*] Convergent Power Series
[*] Relations Between Formal and Convergent Series
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[*] Sums and Products
[*] Quotients
[*] Composition of Series
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[*] Analytic Functions
[*] Differentiation of Power Series
[*] The Inverse and Open Mapping Theorems
[*] The Local Maximum Modulus Principle
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[*] Cauchy's Theorem, First Part
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[*] Holomorphic Functions on Connected Sets
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[*] Appendix: Connectedness
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[*] Integrals Over Paths
[*] Local Primitive for a Holomorphic Function
[*] Another Description of the Integral Along a Path
[*] The Homotopy Form of Cauchy's Theorem
[*] Existence of Global Primitives. Definition of the Logarithm
[*] The Local Cauchy Formula
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[*] Winding Numbers and Cauchy's Theorem
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[*] The Winding Number
[*] The Global Cauchy Theorem
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[*] Dixon's Proof of Theorem 2.5 (Cauchy's Formula)
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[*] Artin's Proof
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[*] Applications of Cauchy's Integral Formula
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[*] Uniform Limits of Analytic Functions
[*] Laurent Series
[*] Isolated Singularities
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[*] Removable Singularities
[*] Poles
[*] Essential Singularities
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[*] Calculus of Residues
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[*] The Residue Formula
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[*] Residues of Differentials
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[*] Evaluation of Definite Integrals
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[*] Fourier Transforms
[*] Trigonometric Integrals
[*] Mellin Transforms
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[*] Conformal Mappings
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[*] Schwarz Lemma
[*] Analytic Automorphisms of the Disc
[*] The Upper Half Plane
[*] Other Examples
[*] Fractional Linear Transformations
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[*] Harmonic Functions
[LIST]
[*] Definition
[LIST]
[*] Application: Perpendicularity
[*] Application: Flow Lines
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[*] Examples
[*] Basic Properties of Harmonic Functions
[*] The Poisson Formula
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[*]
The Poisson Integral as a Convolution
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[*] Construction of Harmonic Functions
[*] Appendix. Differentiating Under the Integral Sign
[/LIST]
[/LIST]
[*] Geometric Function Theory
[LIST]
[*] Schwarz Reflection
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[*] Schwarz Reflection (by Complex Conjugation)
[*] Reflection Across Analytic Arcs
[*] Application of Schwarz Reflection
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[*] The Riemann Mapping Theorem
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[*] Statement of the Theorem
[*] Compact Sets in Function Spaces
[*] Proof of the Riemann Mapping Theorem
[*] Behavior at the Boundary
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[*] Analytic Continuation Along Curves
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[*] Continuation Along a Curve
[*] The Dilogarithm
[*] Application to Picard's Theorem
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[/LIST]
[*] Various Analytic Topics
[LIST]
[*] Applications of the Maximum Modulus Principle and Jensen's Formula
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[*] Jensen's Formula
[*] The Picard-Borel Theorem
[*] Bounds by the Real Part, Borel-Caratheodory Theorem
[*] The Use of Three Circles and the Effect of Small Derivatives
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[*] Hermite Interpolation Formula
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[*] Entire Functions with Rational Values
[*] The Phragmen-Lindelof and Hadamard Theorems
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[*] Entire and Meromorphic Functions
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[*] Infinite Products
[*] Weierstrass Products
[*] Functions of Finite Order
[*] Meromorphic Functions, Mittag-Leffler Theorem
[/LIST]
[*] Elliptic Functions
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[*] The Liouville Theorems
[*] The Weierstrass Function
[*] The Addition Theorem
[*] The Sigma and Zeta Functions
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[*] The Gamma and Zeta Functions
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[*] The Differentiation Lemma
[*] The Gamma Function
[LIST]
[*] Weierstrass Product
[*] The Gauss Multiplication Formula (Distribution Relation)
[*] The (Other) Gauss Formula
[*] The Mellin Transform
[*] The Stirling Formula
[*] Proof of Stirling's Formula
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[*] The Lerch Formula
[*] Zeta Functions
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[*] The Prime Number Theorem
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[*] Basic Analytic Properties of the Zeta Function
[*] The Main Lemma and its Application
[*] Proof of the Main Lemma
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[/LIST]
[*] Appendix
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[*] Summation by Parts and Non-Absolute Convergence
[*] Difference Equations
[*] Analytic Differential Equations
[*] Fixed Points of a Fractional Linear Transformation
[*] Cauchy's Formula for C^\infty Functions
[*] Cauchy's Theorem for Locally Integrable Vector Fields
[*] More on Cauchy-Riemann
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[*] Bibliography
[*] Index
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