Hi,

This is the page for the Complex Variable Module of EE2007 to be offered in Aug 2006 semester.

In this module, we will go through the following topics:

  • Complex numbers and complex functions, Calculus of Complex functions 
  • Integration in the complex plane, 
  • Cauchy integral theorem, evaluation of real integrals.

And here is a mind map (posted Aug05) summarizing the main points in this module which may be useful for revision purpose.

Course Document.

A/Prof LING Keck Voon

E-mail: ekvling@ntu.edu.sg

Tel: 6790 5567

Office: S2-B2A-22

 

Pre-requisite. It is assumed that you have some exposure to elementary complex numbers, i.e. you know something about rectangular and polar representation of complex numbers, addition and multiplication of two complex numbers. 

Self-assessment Package. However, from my July 2002 experience, the above pre-requisite assumption may not be true. So, here is a short quiz for you. The quiz is from Paul Scott of Adelaide University (see below). If you find that there is a knowledge gap, then you should do something to close this gap. For example, you can follow this self-study package from the Mathematics Support Materials (Plymouth University) or other self-study links provided below. Plymouth University also has other self-study materials on the web which might be useful if you are weak in Maths.

Self-study Materials on the Web

·         "Just the Maths!" - a comprehensive set of lecture notes for use on engineering courses, compiled by Tony Hobson of Coventry University. For those who need a revision on Complex Numbers, take a look at units 6.1 to 6.6. These are in PDF format, 7 to 8 pages per unit, with exercises and answers, good for self-study. The units may be freely downloaded but please acknowledge their source.

·         Alternatively, you may try the "MathHelp Notebook on Complex Numbers" from the Geomaths Project, University College, London.

·         Professor Paul Scott of Adelaide University has a Complex Analysis course on the Web, in both HTML and PDF versions. Although his course covers much more depth and details than what we would here, it serves as a very good reference if you want to find out more about this subject.  A very good feature of this site is the numerous examples and quizzes provided for students to gauge their progress. Here I have selected a few quizzes from Paul Scott's site relevant for this course:

o        Quizzes on Complex Numbers, Concept of Continuity, Complex Differentiation, Cauchy-Riemann Equations, Line Integration, and Cauchy Integral.

·         "OpenCourseWare" from MIT includes an online courses in Complex Variables with Applications. Slightly more advanced than the "Just the Maths" materials.

Interesting Stories and Applications of Complex Numbers and Complex Analysis

·         Here, you will find the origin of complex numbers (not from solving quadratic but from cubic equations), and how complex numbers can be used to generate beautiful pictures such as JULIA and MANDELBROT SETS, etc. The site contains a complete set of Maple lessons for an undergraduate course in Complex Analysis or Complex Variables developed by Dr. John Mathews of the California State University Fullerton as part of his book, Complex Analysis: for Mathematics & Engineering, 4th Ed, 2001, co-authored with Dr. Russell Howell of Westmont College.

·         In the article "The Colors of an Equation's Roots" (Science News, April 19, 2003), Ivars Peterson described how a computer scientist Bahman Kalantari of Rutgers University has turned the process of approximating the roots of polynomials involving complex numbers into a method of generating aesthetically pleasing patterns and intriguing artworks.

·         Professor Douglas N. Arnold at the University of Minnesota has a collection of graphical demonstration of concepts in complex analysis. Outside your syllabus but for those interested, check it out to see some interesting graphical illustrations of complex analysis such as these shown here:

The squaring mapExponential, sine, and cosineConformalityMöbius transformationsSchwarz-Christoffel mapsLinks 

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[VCA cover illustration]

My favourite complex analysis book is the Visual Complex Analysis by Tristan Needham. It brings the subject to life by using geometry (not calculation) as the means of explanation. It contains over 500 diagrams to illuminate the geometric reasoning. The book is a bit advanced but should be accessible to students with stronger mathematics background.
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An Imaginary Tale: The Story of i [the square root of minus one]

Paul J. Nahin, Princeton University Press, 1998.

Cloth | 1998 | $29.95 / £18.95 | ISBN: 0-691-02795-1
274 pp. | 6 x 9 | 47 line illus. 1 halftone

An interesting book about the story of complex numbers, almost like reading a novel, not a mathematics book. See http://www.pupress.princeton.edu/titles/6388.html.

 
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For those young at heart, this is a children’s book-on-the-Web from Australia, designed to introduce complex numbers in story form in a way that is intuitive and enjoyable for students.

See http://mathforum.org/johnandbetty/

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