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:
- Brown, J.W. & Churchill, R.V. (9th Ed., 2014). Complex variables and applications, McGraw-Hill
- Spiegel, M., Lipschutz, S. & Schiller, J. (2nd Ed., 2009). Schaum’s Outline of Complex Variables, McGraw-Hill
- Hann, L. & Epstein, B. (1st Ed., 1996). Classical Complex Analysis, Jones and Bartlett Publishers
- Ponnusamy, S. (2nd Ed., 2005). Foundation of Complex Analysis, Alpha Science