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| Module Availability |
| Spring |
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| Assessment Pattern |
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Unit(s) of Assessment
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Weighting Towards Module Mark( %)
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Coursework
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25
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Written examination (2 hours, unseen)
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75
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Qualifying Condition(s)
An overall aggregate mark of 40% for the module.
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| Module Overview |
| Complex analysis, traditionally known as the theory of functions of a complex variable, is one of the fundaments of modern mathematics and used in many branches of mathematics and physics. In this module the main results of this theory are presented. |
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| Prerequisites/Co-requisites |
| CALCULUS MAT1015 |
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| Module Aims |
| This module introduces students to the theory of functions of a complex variable. |
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| Learning Outcomes |
At the end of the module the student should have a thorough understanding of the theory of complex functions of a complex variable and should be able to apply this knowledge in a variety of contexts. In particular the student should be able to:
* Understand what complex differentiation is.
* Quote, derive and apply the Cauchy-Riemann equations.
* Study convergence properties of a complex power series
* Perform contour integrations of continuous functions
* Understand and apply Cauchy's theorem and Cauchy's formula
* Derive analyticity and Liouville's theorem from Cauchy's formula
* Apply Taylor's and Laurent's theorems to obtain power series expansions
* Identify and classify singularities of complex functions and find residues of simple
poles.
* Derive and apply the residue theorem to compute real integrals using residue calculus |
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| Module Content |
| Complex functions, complex differentiability; Cauchy-Riemann equations and harmonic functions; conformal mappings; complex integration: derivation of Cauchy's theorem and Cauchy integration formula, derivation of Liouville theorem and fundamental theorem of algebra; application of Taylor expansion and Laurent expansion; classification of singularities; proof of residue theorem; application to contour integration and evaluation of real integrals. |
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| Methods of Teaching/Learning |
Teaching is by lectures, tutorials and example classes. Learning takes place through lectures, exercises (example sheets), preparation for tests and background reading.
There will be 3 contact hours for 10 weeks consisting of lectures, tutorials and example classes. |
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| Selected Texts/Journals |
Recommended:
- I. Stewart and D. Tall, Complex analysis, CUP, (1987).
- M.R. Spiegel, Complex Variables, Schaum's outline, McGraw Hill, (1981).
Further
Reading
:
- J. Marsden and M. Hoffman, Basic Complex Analysis, Freeman & co (1998). |
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| Last Updated |
| September 10 |
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