**Undergraduate
Modules**

Chapter 1. Complex Numbers

- The Origin of Complex Numbers
- The Algebra of Complex Numbers
- The Geometry of Complex Numbers
- The Geometry of Complex Numbers, Continued
- The Algebra of Complex Numbers, Revisited
- The Topology of Complex Numbers

Chapter 2. Complex Functions

- Complex Functions and Linear Mappings
- The Mappings
z^n andz^1/n- Complex Limits and Continuity
- Branches of Complex Functions
- The Reciprocal Transformation
1/z

Chapter 3. Analytic and Harmonic Functions

- Differentiable and Analytic Functions
- The Cauchy-Riemann Equations
- Harmonic Functions and their Riemann Sheets

Chapter 4. Sequences, Series, and Julia and Mandelbrot Sets

- Complex Sequences and Series
- Julia and Mandelbrot Sets
- Geometric Series and Convergence Theorems
- Power Series Functions

Chapter 5. Elementary Functions

- The Complex Exponential Function
- The Complex Logarithm Function
- Complex Exponents and Powers
- Trigonometric and Hyperbolic Functions
- Inverse Trigonometric and Hyperbolic Functions

Chapter 6. Complex Integration

- Complex Integrals
- Contours and Contour Integrals
- The Cauchy-Goursat Theorem
- The Fundamental Theorem of Integration
- Integral Representations for Analytic Functions
- The Theorems of Morera and Liouville and Applications
- The Fundamental Theorem of Algebra

Chapter 7. Taylor and Laurent Series

- Uniform Convergence
- Taylor Series Representations
- Laurent Series Representations
- Singularities, Zeros and Poles
- Applications of Taylor and Laurent Series

Chapter 8. Residue Theory

- The Residue Theorem
- Trigonometric Integrals
- Improper Integrals of Rational Functions
- Improper Integrals Involving Trigonometric Functions
- Indented Contour Integrals
- Integrands with Branch Points
- The Argument Principle and Rouche's Theorem

Chapter 9. The z-Transforms and Applications

Chapter 10. Conformal Mapping

- Basic Properties of Conformal Mappings
- Mobius Transformations - Bilinear Transformations
- Mappings Involving Elementary Functions
- Mappings by Trigonometric Functions
- Conformal Mapping Dictionary - Part I
- Conformal Mapping Dictionary - Part II
- Conformal Mapping Dictionary - Part III
- Conformal Mapping Dictionary - Part IV
- Conformal Mapping Dictionary - Part V

Chapter 11. Applications of Harmonic Functions

- Preliminaries
- Invariance of Laplace's Equation and the Dirichlet Problem
- Poisson's Integral Formula for the Upper Half Plane
- Two-Dimensional Mathematical Models
- Steady State Temperatures
- Two-Dimensional Electrostatics
- Two-Dimensional Fluid Flow
- The Joukowski Airfoil
- The Schwarz-Christoffel Transformation
- Image of a Fluid Flow
- Sources and Sinks

Chapter 12. Fourier Series and the Laplace Transform

- Fourier Series
- The Dirichlet Problem for the Unit Disk
- Vibrations in Mechanical Systems
- The Fourier Transform
- The Laplace Transform
- Laplace Transforms of Derivatives and Integrals
- Shifting Theorems and the Step Function
- Multiplication and Division by t
- Inverting the Laplace Transform
- Convolution

Return to the Complex Analysis Project

**History of the
Undergraduate Modules**

The undergraduate modules were started in 2003, and since that early
beginning it has been replaced

several times. The current version is under continuous upgrading and
improvement. If you are one to notice

all the details then you probably will find some remnants dated 2003
and the most current version of items

dated 2012. But you need not worry about the time line because the
core material in complex analysis has not

changed much in the past sixty years and some currently available
books have actually made their 60^{th} year

milestone. We cannot brag to have such longevity and are just
thankful that our textbook has just achieved it's

30^{th} year milestone. There have been significant
improvements in the textbook since 1982. Noteworthy is the

new Chapter 9: The
Z-transform
and it's applications to Difference
Equations and
Digital
Signal Filters.

We try to keep
things up to date and this complex analysis web site is one way to do
it. You will find

several instances where the content of the web site goes
significantly beyond the material in the textbook,

e. g. Harmonic
Functions and their Riemann Sheets
in Section
3.3 , and the
new 3-D graphical visualizations

for the residue calculus involving Trigonometric
Integrals ,
Rational
Functions,
Improper
Trig. Integrals,

Indented
Contours, and
Branch
Points. This
is intentional since it allows us to present more details for the

solutions to the examples and exercises. Also you will notice that we
have illustrated how to use Mathematica

and Maple as a pedagogical tool for teaching and exploring concepts
the in complex analysis. Although these

details are much too extensive to print in any textbook, they are
easy to squeeze in on the web pages.

For certain we can say that our book is the first to have included
Maple^{TM }and Mathematica^{TM} supplements.

You are welcome to
correspond with us on matters regarding the content and any
suggestions you have

or typos you may find. You are welcome to correspond with
us by mail or e-mail.

Prof.
John H. Mathews

Department
of Mathematics

California
State University Fullerton

Fullerton,
CA 92634

mathews@fullerton.edu

Prof.
Russell W. Howell

Mathematics
& Computer Science Department

Westmont College

Santa
Barbara, CA 93108

howell@westmont.edu

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

(c) 2012 John H. Mathews, Russell W. Howell