MAS2705 : Complex Analysis & Algebra (Inactive)
- Inactive for Year: 2022/23
- Available for Study Abroad and Exchange students, subject to proof of pre-requisite knowledge.
- Module Leader(s): Dr Zinaida Lykova
- Lecturer: Dr David Stewart
- Owning School: Mathematics, Statistics and Physics
- Teaching Location: Newcastle City Campus
Semesters
Semester 1 Credit Value: | 10 |
Semester 2 Credit Value: | 10 |
ECTS Credits: | 10.0 |
Aims
To introduce the main results in the theory of analytic functions.
To introduce the viewpoint of modern algebra by extending the work of earlier modules and to provide a basis for further study of algebra.
Module summary
The idea of solving an equation like x^2+1 = 0, by inventing a new number to represent the square root of -1, can be traced back at least as far as Egyptian pyramid builders in 1850 BC. From the need to solve such problems the idea of complex numbers, z = x + iy, arose, and happily the rich geometric and analytic theory of these numbers allows simple and elegant solutions to otherwise difficult or insoluble problems.
The important and exciting ideas emerge when we consider functions f(z) of a complex variable z. These functions turn out to have surprising and dramatic properties that are quite unexpected, when compared with functions of real variables (the case when z is a real number). Indeed the natural view
of many common functions, for example exponential and trigonometric functions, is in the complex plane. We shall introduce elementary functions in the complex plane, study their continuity and differentiability, and then move to the extraordinary results that follow when we investigate integration.
Integration in the complex plane involves the notion of a line integral, but surprisingly, many calculations can be reduced to a simple algebraic exercise; something which is quite counter-intuitive, and a great relief to those accustomed to conventional integration of real functions! Indeed, these methods can be used to tackle, very simply, many integrals that are difficult if approached using standard methods.
The theory of complex variables, plays a crucial role in many branches of mathematics and science; for example, in algebraic geometry, number theory, complex dynamics and fractals, fluid mechanics, string theory and electrical engineering.
Modern algebra deals with sets equipped with operations resembling some or all of addition, subtraction, multiplication and division. One example of such a system is a field. The concept of a field is a natural one since we have a familiar structure with two operations which mimic addition and multiplication of numbers. We study examples of finite fields. We consider polynomials defined over a field and means of determining reducibility or irreducibility in some cases. We find that polynomials over a field form a system known as a ring and we consider the possibility of factorisation in a ring. In particular we study rings consisting of complex numbers and see examples of such rings in which some numbers have more than one irreducible factorisation.
Outline Of Syllabus
Complex plane, open and closed sets, limits and continuity. Differentiability: Cauchy-Riemann relations; analytic functions; principal value. Contour integration; Cauchy’s theorem; Cauchy’s integral formula. Poles, residues, Laurent series. Cauchy’s residue theorem. Evaluation of real integrals. Laplace and Fourier transforms: evaluation, inversion.
Multiple examples of rings and fields, including polynomial rings, Zn, and complex number rings. Applications to the study of polynomials, their roots and their factorisation. Factorisation in rings.
Teaching Methods
Teaching Activities
Category | Activity | Number | Length | Student Hours | Comment |
---|---|---|---|---|---|
Scheduled Learning And Teaching Activities | Lecture | 40 | 1:00 | 40:00 | Formal Lectures – Present in Person |
Scheduled Learning And Teaching Activities | Lecture | 4 | 1:00 | 4:00 | Revision Lectures – Present in Person |
Scheduled Learning And Teaching Activities | Lecture | 10 | 1:00 | 10:00 | Problem Classes – Synchronous On-Line |
Guided Independent Study | Assessment preparation and completion | 30 | 1:00 | 30:00 | Completion of in course assessments |
Scheduled Learning And Teaching Activities | Drop-in/surgery | 10 | 1:00 | 10:00 | Synchronous On-Line |
Guided Independent Study | Independent study | 106 | 1:00 | 106:00 | Preparation time for lectures, background reading, coursework review |
Total | 200:00 |
Teaching Rationale And Relationship
Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.
Assessment Methods
The format of resits will be determined by the Board of Examiners
Exams
Description | Length | Semester | When Set | Percentage | Comment |
---|---|---|---|---|---|
Written Examination | 150 | 2 | A | 80 | N/A |
Other Assessment
Description | Semester | When Set | Percentage | Comment |
---|---|---|---|---|
Prob solv exercises | 1 | M | 10 | Problem-solving exercises |
Prob solv exercises | 2 | M | 10 | Problem-solving exercises |
Assessment Rationale And Relationship
A substantial formal unseen examination is appropriate for the assessment of the material in this module. The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.
In the event of on-campus examinations not being possible, an on-line alternative assessment will be used for written examination 1.
Reading Lists
Timetable
- Timetable Website: www.ncl.ac.uk/timetable/
- MAS2705's Timetable