PMAT 31303: Complex Variables

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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
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