Real and Complex Variable
This module runs in alternate years: 2012-13, 2014-15 and so on
Prerequisites: Proof and Structure in Mathematics (or equivalent)
BUEM008S6 (30 credits)
Aims
This module provides the theoretical background calculus, formally defining the idea of a limit, continuity and differentiability of a function and the Riemann integral. The second half of the module extends these ideas to the complex plane. It develops further the need for formal proof in mathematics and the understanding of abstract mathematics.
Teaching and Assessment
Teaching for this module will take place throughout the year, with eight evenings of lectures in each of the Autumn and Spring Terms and two evenings of revision and consolidation in the Summer Term.
Of the final course mark, 80% is based on a three-hour exam in the summer term and the other 20% is from assessed coursework.
Coursework will consist of short, problem based assignments. You will have around three weeks to complete each one.
The examination in the summer term will consist of 8 short (5 mark) questions, which are compulsory, and 4 long (20 mark) questions from which candidates must answer two.
Syllabus
Real Variable
- Number systems: real numbers; supremum and infimum; epsilon notation.
- Sequences: divergence and convergence; limits; subsequences.
- Series: divergence and convergence; tests for divergence and convergence.
- Functions: limits; continuity; tests for continuity and discontinuity.
- Differentiation: Rolle’s Theorem; Mean Value Theorem; l’Hôpital’s rule.
- Integration: Riemann integral; lower and upper Riemann sums.
- Power Series: Taylor polynomial; Taylor’s Theorem.
Complex Variable
- Complex numbers: real and complex parts, modulus, argument, geometrical interpretations.
- Sequences and series: relationships between complex and real sequences, Cauchy sequences, convergent and absolutes convergent series.
- Topology of the complex plane: open and closed sets, properties of open and closed sets, convergent sequences in open and closed sets.
- Complex functions: continuity, boundedness, power series, radius of convergence, ratio and root tests, the exponential function.
- Differentiation: Cauchy-Riemann equations, functions defined by power series, holomorphic functions.
- Integration: Integration of a function along a path, Cauchy’s Theorem, Cauchy’s integral formula, results on integrating holomorphic functions, entire functions, Liouville’s Theorem and its application to the roots of polynomials, Singularities.
- Laurent Series: Computing Laurent series, the residue theorem, calculating residues of functions with simple poles, using the residue theorem to evaluate real integrals.
Learning Outcomes
Subject Specific
- Knowledge and understanding of, and the ability to use, mathematical and/or statistical techniques.
- Knowledge and understanding of a range of results in mathematics.
- Appreciation of the need for proof in mathematics, and the ability to follow and construct mathematical arguments.
- Appreciation of the power of generalization and abstraction in the development of mathematical theories.
- A deeper knowledge of some particular areas of mathematics.
- Appreciation of the historical and cultural aspects of mathematics.
Intellectual
- Ability to comprehend conceptual and abstract material.
- Develop a logical and systematic approach to problem solving.
Practical
- Problem-solving skills, including the ability to assess problems logically and to approach them analytically.
- Highly developed quantitative skills.
- Ability to transfer knowledge and expertise from one context to another.
Personal and Social
- Ability to learn independently using a variety of media.
- Ability to work independently with patience and persistence.
- Time-management and organizational skills.
- Good communication skills, including the ability to write coherently.
- Ability to complete a sustained and substantial task.
- Ability to complete work in a limited time period.