MATH2791: Complex Analysis II

• One of:
• Calculus I (Maths Hons) (MATH1081) OR Calculus I (MATH1061)
• AND:
• Linear Algebra I (Maths Hons) (MATH1091) OR Linear Algebra I (MATH1071)
• AND:
• Analysis I (MATH1051) [may be taken as a co-requisite]

• Analysis 1 (MATH1051) if not taken at Level 1.

• Mathematical Methods in Physics (PHYS2611)

• To introduce the student to the theory of complex analysis.

• The Complex Plane and Riemann Sphere: Complex functions, exponential & logarithm functions (cuts and branches), the Riemann Sphere and extended complex plane.
• Metric spaces: Examples of metric spaces, open/closed sets, convergence, continuity, (path)-connectedness. (Sequential) compactness, Heine-Borel theorem, compactness and continuity.
• Complex differentiation: Complex differentiation, Cauchy-Riemann equations. Connected sets. Conformal maps, holomorphic functions.
• Mbius transformations: Mbius transformations, group law, point at infinity, geometry, circles/lines to circles/lines, preservation of cross ratio, Mbius transformations of unit circle to itself, Riemann sphere and stereographic projection. Mbius transformations as isometries. Biholomorphisms.
• Convergence and Power series: Pointwise and uniform convergence of sequences and series, locally uniform convergence, continuity of the limit, Weierstrass M-test for continuous functions. Power series: Review from Analysis I of disk of (locally uniform) convergence and ratio/root test, term by term differentiation and integration, Taylor series.
• Contour integrals: Curves in the complex plane, line/contour integrals in . Primitives and integrability, the Complex Fundamental Theorem of Calculus, Cauchy's theorem, Cauchy's integral formula, (Cauchy-)Taylor theorem, Morera's theorem.
• Fundamental theorems for holomorphic functions: Liouville's theorem, max mod principle, analytic continuation, the Identity Theorem, harmonic functions.
• Algebraic Topology: Winding numbers, simply connected sets, cycles, Jordan Curve Theorem, general forms of Cauchys Theorem and Cauchys integral formula.
• Singularities and meromorphic functions: Laurent series, singularities, Casorati-Weierstrass, Big Picard Theorem. Cauchys Residue Theorem, Argument Principle, Rouch's Theorem, Open Mapping Theorem.
• Calculus of residues - applications: Evaluation of integrals by calculus of residues, rational functions, rational functions of sine and cosine, indented contours and branch cuts.

Subject-specific Knowledge:

• By the end of the module students will:
• Be able to solve seen and unseen problems on the given topics.
• Be able to reproduce theoretical mathematics in the field of Complex Analysis.
• Have a knowledge and understanding of fundamental theories of these subjects demonstrated through one or more of the following topic areas: Complex Differentiation. Conformal Mappings. Metric Spaces. Contour integrals, calculus of residues. Series, Uniform Convergence. Applications of Complex analysis.

Subject-specific Skills:

• In addition students will have the ability to undertake and defend the use of alternative mathematical skills in the following areas with minimal guidance: Abstract reasoning.

Key Skills:

• Students will have basic mathematical skills in the following areas: abstract reasoning, problem solving.

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Problem classes show how to solve example problems in an ideal way, revealing also the thought processes behind such solutions.
• Tutorials provide active problem-solving engagement and immediate feedback to the learning process.
• Formative assessments provide feedback to guide students in the correct development of their knowledge and skills in preparation for the summative assessment.
• Summative assignments test achievement of learning outcomes and provide feedback to students about their mastery of the topics.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

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