Module MATH2791: Complex Analysis II

Department: Mathematical Sciences

MATH2791: Complex Analysis II

Prerequisites

• 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]

Corequisites

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

Excluded Combination of Modules

• Mathematical Methods in Physics (PHYS2611)

Aims

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

Content

• 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.
• Möbius transformations: Möbius transformations, group law, point at infinity, geometry, circles/lines to circles/lines, preservation of cross ratio, Möbius transformations of unit circle to itself, Riemann sphere and stereographic projection. Möbius 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 ℂ. 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 Cauchy’s Theorem and Cauchy’s integral formula.
• Singularities and meromorphic functions: Laurent series, singularities, Casorati-Weier-strass, Big Picard Theorem. Cauchy’s Residue Theorem, Argument Principle, Rouché's Theo-rem, 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.

Learning Outcomes

• 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.

• 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.

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

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• 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.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

There will be at most one formative or summative assignment each week.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University