PMAT 31303: Complex Variables

Course Code            : PMAT 31303

Title                          : Complex Variables

Pre-requisites          : PMAT 22293

Learning Outcomes:

At the end of this course, the student should be able to

• Perform basic mathematical operations with complex numbers in cartesian and polar form

• demonstrate knowledge of complex numbers and complex valued functions

• analyze and discuss limit, continuity and differentiability of complex valued functions

• analyze and interpret results of complex numbers and complex valued functions in applications

• evaluate real integrals using complex integrals.

Course Contents:

Complex Numbers: 

• Basic Algebraic Properties, Exponential Form, De Movers’ Theorem, nth root of unity, Roots of Complex Numbers, Argand Diagram, Regions in the Complex Plane.

Analytic Functions: 

• Complex Valued Functions, Limits, Theorems on Limits, Continuity and Uniform continuity, Derivatives, Differentiation Formulas, Cauchy–Riemann Equations, Sufficient Conditions for Differentiability, Analytic Functions, Harmonic Functions.

Elementary functions: 

• Polynomial functions, rational functions, exponentials, trigonometric functions, hyperbolic functions, logarithmic functions, power series.

Integrals: 

• Definite Integrals of Complex-Valued Function of a Complex Variable, Contour Integrals, Properties of integrals, Cauchy Theorem, Cauchy Integral Formula.

Series: 

• Taylor Series, Laurent Series, Classification of Singular Points.

Residues and Poles: 

• Residues, Residues at Poles, Cauchy’s Residue Theorem, Residue at Infinity.

Applications of Residues: 

• Evaluation of Real Integrals.

Method of Teaching and Learning : A combination of lectures and tutorial discussions.

Assessment : Based on tutorials, tests and end of course examination.

Recommended Reading:

1. Brown, J.W. & Churchill, R.V. (9th Ed., 2014). Complex variables and applications, McGraw-Hill

2. Spiegel, M., Lipschutz, S. & Schiller, J. (2nd Ed., 2009). Schaum’s Outline of Complex Variables, McGraw-Hill

3. Hann, L. & Epstein, B. (1st Ed., 1996). Classical Complex Analysis, Jones and Bartlett Publishers

4. Ponnusamy, S. (2nd Ed., 2005). Foundation of Complex Analysis, Alpha Science