Theory of Functions of Complex
Variables
Lecture Notes
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Contents
1 The Complex Number System and its Properties
1.1 The Complex Plane
1.2 The Triangle Inequality
1.3 Polar Coordinates
1.4 Powers and Roots
1.5 The Exponential Representation
1.6 Lines
1.7 Practice Problems
2 Elementary Theory of Power series
2.1 Sequences
2.2 Complex Series
2.3 Uniform Convergence
2.4 Power Series
2.5 Practice Problems
3 Elementary Point Set Topology of \(\mathbb {C}\)
3.1 Generating Open Sets
3.2 Closed sets
3.3 Compactness
3.4 Connectedness
3.5 Sequences
3.6 Practice Problems
4 Complex Functions
4.1 Functions
4.2 Limits of Complex Functions
4.3 Continuity of Complex Functions
4.4 Differentiation of a Complex Function
4.5 Analytic Functions
4.6 Laplace’s Equations
4.7 Cauchy - Riemann Equations
4.8 Harmonic Functions
4.9 Elementary Functions
4.10 The Logarithm Function
4.11 Trigonometric and Hyperbolic Functions
4.12 Practice Problems
5 Analytic Functions as Mappings
5.1 Mappings
5.2 Linear Transformation
5.3 The Identity Mapping
5.4 The Mapping \(w = z^c\)
5.5 The Mapping \(w = z^{1/2}\)
5.6 The Inversion Mapping \(w = \dfrac {1}{z}\)
5.7 The Point at Infinity
5.8 Fixed Points
5.9 Paths and Smooth Paths
5.10 Linear Fractional Transformations
5.11 Practice Problems
6 Complex Integration
6.1 Arc, Curve, Contour
6.2 Contour Integration
6.3 The Cauchy - Goursat Theorem
6.4 Antiderivatives
6.5 Definite Integral
6.6 The Cauchy Integral Formula
6.7 Counting Zeroes
6.8 Practice Problems
7 The Maximum Modulus Theorem
7.1 The Maximum Modulus Principle
7.2 Morera’s Theorem
7.3 Liouville’s Theorem
7.4 The Fundamental Theorem of Algebra
7.5 Cauchy Inequality
7.6 Schwarz Lemma
7.7 Practice Problems
8 Improper Integrals Using Contour Integration
8.1 Jordan Inequality
8.2 Practice Problems
9 Calculus of Residues
9.1 Series
9.2 Singularities and Zeroes
9.3 Residues and the Residue Theorem
9.4 The Argument Principle
9.5 Practice Problems
10 Analytic Continuation
10.1 Uniqueness of the Continuation
10.2 Continuation Along a Path
10.3 The Logarithm, and Why a Function Can Fail to be Single-Valued
10.4 Homotopy and the Monodromy Theorem
10.5 The Schwarz Reflection Principle
10.6 When Continuation is Impossible: Natural Boundaries
10.7 Practice Problems
11 Problems Beyond These Notes
11.1 The Graduate Syllabus Behind These Problems
References
References
1 The Complex Number System and its Properties
1.1 The Complex Plane
1.2 The Triangle Inequality
1.3 Polar Coordinates
1.4 Powers and Roots
1.5 The Exponential Representation
1.6 Lines
1.7 Practice Problems
2 Elementary Theory of Power series
2.1 Sequences
2.2 Complex Series
2.3 Uniform Convergence
2.4 Power Series
2.5 Practice Problems
3 Elementary Point Set Topology of \(\mathbb {C}\)
3.1 Generating Open Sets
3.2 Closed sets
3.3 Compactness
3.4 Connectedness
3.5 Sequences
3.6 Practice Problems
4 Complex Functions
4.1 Functions
4.2 Limits of Complex Functions
4.3 Continuity of Complex Functions
4.4 Differentiation of a Complex Function
4.5 Analytic Functions
4.6 Laplace’s Equations
4.7 Cauchy - Riemann Equations
4.8 Harmonic Functions
4.9 Elementary Functions
4.10 The Logarithm Function
4.11 Trigonometric and Hyperbolic Functions
4.12 Practice Problems
5 Analytic Functions as Mappings
5.1 Mappings
5.2 Linear Transformation
5.3 The Identity Mapping
5.4 The Mapping \(w = z^c\)
5.5 The Mapping \(w = z^{1/2}\)
5.6 The Inversion Mapping \(w = \dfrac {1}{z}\)
5.7 The Point at Infinity
5.8 Fixed Points
5.9 Paths and Smooth Paths
5.10 Linear Fractional Transformations
5.11 Practice Problems
6 Complex Integration
6.1 Arc, Curve, Contour
6.2 Contour Integration
6.3 The Cauchy - Goursat Theorem
6.4 Antiderivatives
6.5 Definite Integral
6.6 The Cauchy Integral Formula
6.7 Counting Zeroes
6.8 Practice Problems
7 The Maximum Modulus Theorem
7.1 The Maximum Modulus Principle
7.2 Morera’s Theorem
7.3 Liouville’s Theorem
7.4 The Fundamental Theorem of Algebra
7.5 Cauchy Inequality
7.6 Schwarz Lemma
7.7 Practice Problems
8 Improper Integrals Using Contour Integration
8.1 Jordan Inequality
8.2 Practice Problems
9 Calculus of Residues
9.1 Series
9.2 Singularities and Zeroes
9.3 Residues and the Residue Theorem
9.4 The Argument Principle
9.5 Practice Problems
10 Analytic Continuation
10.1 Uniqueness of the Continuation
10.2 Continuation Along a Path
10.3 The Logarithm, and Why a Function Can Fail to be Single-Valued
10.4 Homotopy and the Monodromy Theorem
10.5 The Schwarz Reflection Principle
10.6 When Continuation is Impossible: Natural Boundaries
10.7 Practice Problems
11 Problems Beyond These Notes
11.1 The Graduate Syllabus Behind These Problems
References
References