PMAT 31593: Complex Variables

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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:

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