The module aims to provide an understanding of the basic facts of complex analysis, in particular of the nice properties enjoyed by the derivatives and integrals of functions of a complex variable, and to show how complex analysis can be used to evaluate real integrals. The course also aims to provide an understanding of the basic concepts in analysis in the context of metric spaces and to show how these ideas are generalisations of the ideas used in previous moduleMultivariable Calculus and Real Analysis and to improve the students abilities in mathematical reasoning and in expressing themselves accurately in writing by producing correct mathematical proofs.
1.1 Revision of Complex numbers: revision of algebraic operations on complex numbers, the Argand diagram and de Moivre’s theorem, 1.2 , 1.3 Definition of a Metric Spaces, Limits of Sequences and Continuous Functions:definition and examples of metric spaces, sequences and limits, sub-sequences and bounded sequences, continuous functions; formal and sequential definitions., 1.4 , 1.5 Complex Functions Paths and Path integrals :complex functions, paths in the complex plane, path integrals, the exponential, trig, hyperbolic and logfunctions, the index of a closed path at a point., 1.6 , 1.7 Cauchy’s theorem and its consequences: Differentiation of complex functions, the Cauchy Riemann equations, Cauchy’s theorem for a triangle, Cauchy’s theorem for a con-vex set, Cauchy’s integral formula. Cauchy’s formula for Derivatives., 1.8 , 1.9 Complex series: Convergence of series and power series, radius of convergence of power series, Taylor series, Liouville’s theorem, the fundamental theorem of algebra., 1.10 , 1.11 Cauchy Residue Theorem :Zeros and Poles, Laurent series, Residues, Cauchy’s residue theorem, evaluation of real integrals, summation of real series., 1.12 , 1.13 Open, closed sets and compact subsets of a metric spaces:open spheres, open sets, closed sets and closure points in metric spaces, sequential characterisation of closed sets, compact metric spaces and their properties
By the end of the course, students should be able to do the following:
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SCQF Level: 9
Credits: 15