**Course Code ****:** PMAT 31593

**Title ****:** Complex Variables

**Pre****-****requisites ****:** PMAT 22583

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**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

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**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

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

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

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**Recommended Reading****: **

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