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Semester 2
Honours Complex Variables (MATH10067)
Subject
Mathematics
College
SCE
Credits
20
Normal Year Taken
3
Delivery Session Year
2023/2024
Pre-requisites
Visiting students are advised to check that they have studied the material covered in the syllabus of each pre-requisite course before enrolling.
Course Summary
This is a first course in complex analysis. Topics are: analytic functions, Moebius transformations and the Riemann sphere, complex integration, series expansions, the residue calculus and its applications.In the 'skills' section of this course we will work on mathematical reading and writing, although the skills involved are widely applicable to reading and writing technical and non-technical reports. Students will then use these skills to complete a project researching a topic connected with complex numbers or complex analysis and produce a written report.
Course Description
Holomorphic functions: complex numbers, algebraic, geometric and topological properties of the complex plane, functions of a complex variable, differentiability and holomorphicity, Cauchy-Riemann equations, harmonic functions. Examples: polynomials, rational functions, exponential and related functions. Multivalued functions: the logarithm and complex powers, branches and an example of a Riemann surface.Holomorphic functions as mappings: Moebius transformations, the extended complex plane and the Riemann sphere, the cross-ratio. How to visualise functions of a complex variable.Complex integration: contour integrals, independence of path, the Cauchy Integral Theorem, the Cauchy Integral Formulae, Morera's Theorem, Liouville's Theorem and its applications, the Maximum modulus principle.Series expansions: holomorphic functions as analytic functions, Taylor and Laurent series, zeros, singularities, analytic continuation.Residue theory: the Cauchy Residue Theorem, the argument principle, Rouche's Theorem and applications to the evaluation of real integrals and of sums.Skills: The content will be chosen appropriate to the learning outcomes.
Assessment Information
Written Exam 75%, Coursework 25%, Practical Exam 0%
Additional Assessment Information
Coursework 25%, Examination 75%
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