Aims

To introduce the main results in the theory of analytic functions.

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 most fruitful way of looking at common functions (such as the exponential and trigonometric functions) is to study them as functions of a complex variable. 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

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.

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking. 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 Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

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.