Complex Analysis

This subject has the following teaching availabilities in 2010:

One of

• 620-295 Real Analysis with Applications
• 620-158 Accelerated Mathematics 2

And any other second year level subject from the Department of Mathematics and Statistics.

Students may only gain credit for one of

• 620-324 Complex Analysis,
• 620-221 Real and Complex Analysis (prior to 2009)
• 620-252 Analysis (prior to 2010).

Complex analysis is a core subject in pure and applied mathematics, as well as the physical and engineering sciences. While it is true that physical phenomena are given in terms of real numbers and real variables, it is often too difficult and sometimes not possible, to solve the algebraic and differential equations used to model these phenomena without introducing complex numbers and complex variables and applying the powerful techniques of complex analysis.

Topics include: the topology of the complex plane; convergence of complex sequences and series; analytic functions, the Cauchy-Riemann equations, harmonic functions and applications; contour integrals and the Cauchy Integral Theorem; singularities, Laurent series, the Residue Theorem, evaluation of integrals using contour integration, conformal mapping; and aspects of the gamma function.

At the completion of this subject, students should understand the concepts of analytic function and contour integral and should be able to:

• apply the Cauchy-Riemann equations
• use the complex exponential and logarithm
• apply Cauchy’s theorems concerning contour integrals
• apply the residue theorem in a variety of contexts
• understand theoretical implications of Cauchy’s theorems such as the maximum modulus principle, Liouville’s Theorem and the fundamental theorem of algebra

Three or four written assignments due at regular intervals during semester amounting to a total of up to 50 pages (20%), and a 3-hour written examination in the examination period (80%).

Jerrold Marsden and Michael J. Hoffman, Basic Complex Analysis, 3rd Ed. Freeman, 1998.

This subject potentially can be taken as a breadth subject component for the following courses:

• Bachelor of Commerce
• Bachelor of Environments
• Bachelor of Music

You should visit learn more about breadth subjects and read the breadth requirements for your degree, and should discuss your choice with your student adviser, before deciding on your subjects.

In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include:

• problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
• analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
• collaborative skills: the ability to work in a team;
• time-management skills: the ability to meet regular deadlines while balancing competing commitments.