Newcastle University

Pre Requisite Comment

N/A

Co Requisite Comment

N/A

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.

Intended Knowledge Outcomes

Students will know the basic principles underlying the existence and differentiability of functions defined in the complex plane; they will know the 3 fundamental theorems of complex integration and how to use these to evaluate real integrals; they will have seen examples of the Fourier Transform.

Students will develop an understanding of the concepts of rings and fields. They will understand the relationship between polynomials defined over a field and the field itself. They will learn about factorisation theory in rings and see examples where unique factorisation fails.

Intended Skill Outcomes

Students will be able to manipulate functions defined in the complex plane, and be able to test their differentiability; they will be able to perform some types of contour integration and use these techniques for evaluating certain real integrals.

Students will be able to perform calculations over a field. They will be able to test polynomials for reducibility in certain cases and to factorise polynomials over Zp, Q, R, and C into irreducible factors. They will be able to factorise elements of complex number rings into irreducible factors. Students will be able to demonstrate an understanding of the possibility of non-uniqueness of factorisation in rings, as exemplified by certain complex number rings.

Teaching Rationale And Relationship

Non-synchronous online materials are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on assessed work. Present-in-person and synchronous online sessions are used to help develop the students’ abilities at applying the theory to solving problems and to identify and resolve specific queries raised by students, and to allow students to receive individual feedback on marked work. Students who cannot attend a present-in-person session will be provided with an alternative activity allowing them to access the learning outcomes of that session. In addition, office hours/discussion board activity will provide an opportunity for more direct contact between individual students and the lecturer: a typical student might spend a total of one or two hours over the course of the module, either individually or as part of a group.
Alternatives will be offered to students unable to be present-in-person due to the prevailing C-19 circumstances.
Student’s should consult their individual timetable for up-to-date delivery information.

Assessment Rationale And Relationship

A substantial formal examination is appropriate for the assessment of the material in this module. The course assessments will 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.

General Notes

N/A