Complex analysis is a classic branch of mathematics with a long and rich history. It plays a role in many areas of modern mathematics, including geometry, number theory, and dynamical systems, but is also widely used in engineering and physics. Complex analysis extends the fundamental concepts of calculus to complex numbers. You'll explore, in detail, the intricate relationships between functions of a complex variable and the geometry and algebra of the complex plane, which provide a rich theoretical framework with many striking results including, Cauchy's integral formula, Laurent's theorem, and the residue theorem. You'll master new techniques to compute integrals of functions of a real variable using contour integration, see how complex analysis can be used to prove the Fundamental Theorem of Algebra, and gain an appreciation of its wide-ranging applications. Not only will you expand your mathematical toolkit, but also gain a deeper appreciation of the unity of mathematics.
Lecture 48, Revision Session 4,Independent Study 98, Assessment Preparation 50.
Spring
Problem sheets worth 30%.
Examination (2 hours) worth 70%.
Reassessment Method: Like-for-like Including composite form of reassessment for failed portfolio component – written single problem sheet.
On successfully completing the module, students will be able to:
1) Deploy and apply of advanced methods and results in complex analysis, including contour integration, residue calculus, Cauchy’s integral formula, Cauchy-Riemann equations, and power series.
2) Apply accurately established sophisticated computational techniques in complex analysis to solve problems.
3) Deploy theory and core results of complex analysis to devise approaches and strategies to solve problems.
4) Critically evaluate arguments, assumptions, and abstract concepts to make sound judgements, and to formulate appropriate approaches to solve problems within the theory of complex analysis.
5) Plan their own learning, and interpret scholarly texts and literature.
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