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Complex Analysis:
an Open Source Textbook
Russell W. Howell, John H. Mathews
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Front Matter
Preface
1
Complex Numbers
1.1
The Origin of Complex Numbers
1.1.1
Geometric Progress of John Wallis
1.1.2
A Geometric Representation of Real Numbers
1.1.3
A Geometric Representation of Complex Numbers
1.1.4
Caspar Wessel Makes a Breakthrough
1.1.4
Exercises
1.2
The Algebra of Complex Numbers, Part I
1.2
Exercises
1.3
The Geometry of Complex Numbers, Part I
1.3
Exercises
1.4
The Geometry of Complex Numbers, Part II
1.4
Exercises
1.5
The Algebra of Complex Numbers, Part II
1.5
Exercises
1.6
The Topology of Complex Numbers
1.6
Exercises
2
Complex Functions
2.1
Functions and Linear Mappings
2.1
Exercises
2.2
The Mappings
\(w=z^n\)
and
\(w=z^{\frac{1}{n}}\)
2.2
Exercises
2.3
Limits and Continuity
2.3
Exercises
2.4
Branches of Functions
2.4.1
The Riemann Surface for
\(\mathbf{w=z^{\frac{1}{2}}}\)
2.4.1
Exercises
2.5
The Reciprocal Transformation
\(w=\frac{1}{z}\)
2.5
Exercises
3
Analytic and Harmonic Functions
3.1
Differentiable and Analytic Functions
3.1
Exercises
3.2
The Cauchy-Riemann Equations
3.2
Exercises
3.3
Harmonic Functions
3.3
Exercises
4
Sequences, Series, and Fractals
4.1
Sequences and Series
4.1
Exercises
4.2
Julia and Mandelbrot Sets
4.2
Exercises
4.3
Geometric Series and Convergence Theorems
4.3
Exercises
4.4
Power Series Functions
4.4
Exercises
5
Elementary Functions
5.1
The Complex Exponential Function
5.1
Exercises
5.2
The Complex Logarithm
5.2
Exercises
5.3
Complex Exponents
5.3
Exercises
5.4
Trigonometric and Hyperbolic Functions
5.4
Exercises
5.5
Inverse Trigonometric and Hyperbolic Functions
5.5
Exercises
6
Complex Integration
6.1
Complex Integrals
6.1
Exercises
6.2
Contours and Contour Integrals
6.2
Exercises
6.3
The Cauchy-Goursat Theorem
6.3
Exercises
6.4
The Fundamental Theorems of Integration
6.4
Exercises
6.5
Integral Representations
6.5
Exercises
6.6
The Theorems of Morera and Liouville
6.6
Exercises
7
Taylor and Laurent Series
7.1
Uniform Convergence
7.1
Exercises
7.2
Taylor Series Representations
7.2
Exercises
7.3
Laurent Series Representations
7.3
Exercises
7.4
Singularities, Zeros, and Poles
7.4
Exercises
7.5
Applications of Taylor and Laurent Series
7.5
Exercises
8
Residue Theory
8.1
The Residue Theorem
8.1
Exercises
8.2
Trigonometric Integrals
8.2
Exercises
8.3
Improper Integrals of Rational Functions
8.3
Exercises
8.4
Improper Integrals of Trigonometric Functions
8.4
Exercises
8.5
Indented Contour Integrals
8.5
Exercises
8.6
Integrands with Branch Points
8.6
Exercises
8.7
The Argument Principle & Rouché’s Theorem
8.7
Exercises
9
Conformal Mapping
9.1
Basic Properties of Conformal Mappings
9.1
Exercises
9.2
Bilinear Transformations
9.2.1
Lines of Flux
9.2.1
Exercises
9.3
Mappings Involving Elementary Functions
9.3.1
The Mapping
\({\bm w=(z^2-1)^\frac{1}{2}}\)
9.3.2
The Riemann Surface for
\(\bm w=(z^2-1)^\frac{1}{2}\)
9.3.2
Exercises
9.4
Mapping by Trigonometric Functions
9.4.1
The Complex Arcsine Function
9.4.1
Exercises
10
Applications of Harmonic Functions
10.1
Preliminaries
10.2
The Dirichlet Problem
10.2
Exercises
10.3
Poisson’s Integral Formula
10.3
Exercises
10.4
Two-Dimensional Mathematical Models
10.5
Steady State Temperatures
10.5.1
An Insulated Segment on the Boundary
10.5.1
Exercises
10.6
Two-Dimensional Electrostatics
10.6
Exercises
10.7
Two-Dimensional Fluid Flow
10.7
Exercises
10.8
The Joukowski Airfoil
10.8.1
Flow with Circulation
10.8.1
Exercises
10.9
The Schwarz-Christoffel Transformation
10.9
Exercises
10.10
Image of a Fluid Flow
10.10
Exercises
10.11
Sources and Sinks
10.11.1
Source: A Charged Line
10.11.1
Exercises
11
Fourier Series and the Laplace Transform
11.1
Fourier Series
11.1.1
Proof of Euler’s Formulas
11.1.1
Exercises
11.2
The Dirichlet Problem for the Unit Disk
11.2
Exercises
11.3
Vibrations in Mechanical Systems
11.3.1
Damped System
11.3.2
Forced Vibrations
11.3.2
Exercises
11.4
The Fourier Transform
11.4
Exercises
11.5
The Laplace Transform
11.5.1
From Fourier Transforms to Laplace Transforms
11.5.2
Properties of the Laplace Transform
11.5.2
Exercises
11.6
Laplace Transforms of Derivatives and Integrals
11.6
Exercises
11.7
Shifting Theorems and the Step Function
11.7
Exercises
11.8
Multiplication and Division by
\(t\)
11.8
Exercises
11.9
Inverting the Laplace Transform
11.9
Exercises
11.10
Convolution
11.10
Exercises
Complex Analysis:
an Open Source Textbook
Russell W. Howell
Kathleen Smith Professor of Mathematics
Westmont College, Santa Barbara, CA
howell@westmont.edu
John H. Mathews
Professor of Mathematics, Emeritus
California State University, Fullerton, CA
fmathews394@gmail.com
Last Update: January 23, 2025
