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### Course Overview

This course asks what happens when concepts such as convergence of sequences and series, continuity and differentiability, are applied in the complex plane? The results are much more beautiful, and often, surprisingly, simpler, than over the real numbers. This course also covers contour integration of complex functions, which has important applications in Physics and Engineering.

### Course Details

Study Type Level Undergraduate 4 Second Sub Session 15 credits (7.5 ECTS credits) None. No Dr Zur Izhakian

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### Course Description

• Revision of complex numbers, roots of unity, polynomials.
• Elementary functions, differentiation, Cauchy-Riemann equations.
• Path integrals, Cauchy's Theorem and Cauchy's Integral Formulae.
• Liouville's Theorem and the Fundamental Theorem of Algebra.
• Taylor Series, Laurent Series, Cauchy's Residue Theorem and applications to real integrals.

Syllabus

• Complex numbers and complex plane.
• Complex sequences and complex series.
• Complex functions.
• Sets, limits, and continuous functions.
• Derivatives
• Power series.
• Integration
• Cauchy’s theorem.
• Taylor series.
• Laurent series.
• Residues.

### Degree Programmes for which this Course is Prescribed

• BSc Mathematics & Engineering Mathematics
• BSc Mathematics with French
• BSc Mathematics with Gaelic
• BSc Mathematics-Physics
• MA Business Management - Mathematics
• MA Mathematics-Philosophy
• MA Mathematics-Sociology
• Mathematics Joint

### Contact Teaching Time

54 hours

This is the total time spent in lectures, tutorials and other class teaching.

### Teaching Breakdown

• 2 Lectures during University weeks 25 - 35
• 1 Tutorial during University weeks 25 - 35

### Summative Assessments

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%).

Resit: 1 two-hour examination (maximum of 100% resit and 80% resit with 20% in-course assessment).

### Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

### Feedback

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact Course Coordinators for feedback on the final examination.

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