Complex Variables - Complex Analysis 

Undergraduate Modules

Return to the Complex Analysis Project


Chapter 1. Complex Numbers
  1. The Origin of Complex Numbers
  2. The Algebra of Complex Numbers
  3. The Geometry of Complex Numbers
  4. The Geometry of Complex Numbers, Continued
  5. The Algebra of Complex Numbers, Revisited
  6. The Topology of Complex Numbers


Chapter 2. Complex Functions

  1. Complex Functions and Linear Mappings
  2. The Mappings z^n and z^1/n
  3. Complex Limits and Continuity
  4. Branches of Complex Functions
  5. The Reciprocal Transformation 1/z


Chapter 3. Analytic and Harmonic Functions

  1. Differentiable and Analytic Functions
  2. The Cauchy-Riemann Equations
  3. Harmonic Functions and their Riemann Sheets


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

  1. Complex Sequences and Series
  2. Julia and Mandelbrot Sets
  3. Geometric Series and Convergence Theorems
  4. Power Series Functions


Chapter 5. Elementary Functions

  1. The Complex Exponential Function
  2. The Complex Logarithm Function
  3. Complex Exponents and Powers
  4. Trigonometric and Hyperbolic Functions
  5. Inverse Trigonometric and Hyperbolic Functions


Chapter 6. Complex Integration

  1. Complex Integrals
  2. Contours and Contour Integrals
  3. The Cauchy-Goursat Theorem
  4. The Fundamental Theorem of Integration
  5. Integral Representations for Analytic Functions
  6. The Theorems of Morera and Liouville and Applications
  7. The Fundamental Theorem of Algebra


Chapter 7. Taylor and Laurent Series

  1. Uniform Convergence
  2. Taylor Series Representations
  3. Laurent Series Representations
  4. Singularities, Zeros and Poles
  5. Applications of Taylor and Laurent Series


Chapter 8. Residue Theory

  1. The Residue Theorem
  2. Trigonometric Integrals
  3. Improper Integrals of Rational Functions
  4. Improper Integrals Involving Trigonometric Functions
  5. Indented Contour Integrals
  6. Integrands with Branch Points
  7. The Argument Principle and Rouche's Theorem


Chapter 9. The z-Transforms and Applications

  1. The z-transform
  2. Second-Order Homogeneous Difference Equations
  3. Digital Signal Filters


Chapter 10. Conformal Mapping

  1. Basic Properties of Conformal Mappings
  2. Mobius Transformations - Bilinear Transformations
  3. Mappings Involving Elementary Functions
  4. Mappings by Trigonometric Functions
  5. Conformal Mapping Dictionary - Part I
  6. Conformal Mapping Dictionary - Part II
  7. Conformal Mapping Dictionary - Part III
  8. Conformal Mapping Dictionary - Part IV
  9. Conformal Mapping Dictionary - Part V


Chapter 11. Applications of Harmonic Functions

  1. Preliminaries
  2. Invariance of Laplace's Equation and the Dirichlet Problem
  3. Poisson's Integral Formula for the Upper Half Plane
  4. Two-Dimensional Mathematical Models
  5. Steady State Temperatures
  6. Two-Dimensional Electrostatics
  7. Two-Dimensional Fluid Flow
  8. The Joukowski Airfoil
  9. The Schwarz-Christoffel Transformation
  10. Image of a Fluid Flow
  11. Sources and Sinks


Chapter 12. Fourier Series and the Laplace Transform

  1. Fourier Series
  2. The Dirichlet Problem for the Unit Disk
  3. Vibrations in Mechanical Systems
  4. The Fourier Transform
  5. The Laplace Transform
  6. Laplace Transforms of Derivatives and Integrals
  7. Shifting Theorems and the Step Function
  8. Multiplication and Division by t
  9. Inverting the Laplace Transform
  10. 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 60th year

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

30th 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 MapleTM and MathematicaTM 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  

Prof. Russell W. Howell  
                    Mathematics & Computer Science Department  
                    Westmont  College  
                    Santa Barbara,  CA  93108  



















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
























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