Complex Variables - Complex Analysis 

Research Experience for Undergraduates



          This page contains links which are intended to help students find additional resources for

studying various topics in complex analysis. All of the original links were created in 2003 and

were a complementary resource that accompanied the 2001 edition of our textbook.

COMPLEX ANALYSIS: for Mathematics and Engineering, Fourth Edition, 2001
John H. Mathews and Russell W. Howell
ISBN: 0-7637-4125-9
Jones and Bartlett Pub. Inc.
Sudbury, MA


          Since this part of our complex analysis project was created nine years ago, it has been

almost impossible to keep it up to date. However a few of the pages have been updated in

recent years. We apologize for any inconveniences that you might experience with dead links.

It is very time consuming to keep this section of the project up to date. Please be patient.



Complex Numbers
  1. Complex Numbers
  2. DeMoivre's Theorem
  3. Roots of Cubic Equations
  4. Roots of Quartic Equations
  5. Complex Roots of Polynomials
  6. Quaternions
  7. History of Complex Numbers


Complex Functions

  1. Graphics for Complex Functions
  2. Riemann Sphere
  3. Mobius - Bilinear Transformation
  4. Poincaré Disk Model


Analytic and Harmonic Functions

  1. Analytic Functions
  2. Mean Value Theorem and Rolle's Theorem
  3. Cauchy-Riemann Equations
  4. Harmonic Functions
  5. Polya Vector Field
  6. Entire Functions
  7. Holomorphic Functions
  8. Meromorphic Functions


Sequences, Series, and Julia and Mandelbrot Sets

  1. Julia Sets
  2. Mandelbrot Set
  3. Fractals
  4. Geometric Series
  5. Convergence of Series
  6. Power Series


Elementary Functions

  1. Exponential Function
  2. Complex Logarithms
  3. Riemann Surfaces


Complex Integration

  1. Complex Integral
  2. Contour Integrals
  3. Green's Theorem
  4. Cauchy-Goursat Theorem
  5. Cauchy's Integral Formula 
  6. Fundamental Theorem of Calculus
  7. Morera's Theorem
  8. Maximum Modulus Principle
  9. Liouville's Theorem
  10. Fundamental Theorem of Algebra
  11. Schwarz Lemma


Taylor and Laurent Series

  1. Taylor Series
  2. Laurent Series
  3. Poles and Singularity
  4. Infinite Products
  5. Analytic Continuation
  6. Bieberbach Conjecture
  7. Riemann Hypothesis


Residue Theory

  1. Residue Calculus
  2. Contour Integrals
  3. Cauchy Principal Value
  4. Hilbert Transformation
  5. Argument Principle
  6. Rouche's Theorem
  7. Nyquist Stability Criterion
  8. Z-Transform


Conformal Mapping

  1. Conformal Mapping
  2. Smith Chart
  3. Quasiconformal Mapping


Applications of Harmonic Functions

  1. Dirichlet Problem
  2. Neumann Problem
  3. Poisson Integral
  4. Electrostatics
  5. Ideal Fluid Flow
  6. Steady State Temperature
  7. Joukowski Transformation and Airfoils
  8. Schwarz-Christoffel transformation
  9. Complex Potential
  10. Green's Function


Fourier Series and the Laplace Transform

  1. Fourier Series and Transform
  2. Laplace Transform





Return to the Complex Analysis Project



          Finally, we would like to emphasize that the above materials are supplements that are coordinated

with the various editions of our textbook "Complex Analysis for Mathematics and Engineering".

          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