Course Code : PMAT 31593
Title : Complex Variables
Pre-requisites : PMAT 22583
Learning Outcomes:
At the end of this course, the student should be able to
- demonstrate knowledge of complex numbers and complex valued functions
- apply knowledge of complex numbers and complex valued functions in applications
Course Contents:
Complex Numbers: Basic Algebraic Properties, Exponential Form, Roots of Complex Numbers, Regions in the Complex Plane
Analytic Functions: Limits, Theorems on Limits, Continuity, Derivatives, Differentiation Formulas, Cauchy–Riemann Equations, Sufficient Conditions for Differentiability, Analytic Functions, Harmonic Functions
Integrals: Definite Integrals of Complex-Valued Function of a Real Variable, Contour Integrals, Anti-derivatives, Cauchy–Goursat Theorem, Cauchy Integral Formula, Liouville’s Theorem and the Fundamental Theorem of Algebra, Maximum Modulus Principle
Series: Taylor Series, Laurent Series
Residues and Poles: Isolated Singular Points, Residues, Cauchy’s Residue Theorem, Residue at Infinity, Isolated Singular Points, Residues at Poles
Applications of Residues: Evaluation of Improper 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:
- Brown, J.W., Churchill, R.V., (9th edition, 2014) Complex variables and applications, McGraw-Hill