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Module

MAS2702 : Complex Analysis

  • Offered for Year: 2021/22
  • Module Leader(s): Dr Zinaida Lykova
  • Owning School: Mathematics, Statistics and Physics
  • Teaching Location: Newcastle City Campus
Semesters
Semester 1 Credit Value: 10
ECTS Credits: 5.0

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 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.

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 Methods

Teaching Activities
Category Activity Number Length Student Hours Comment
Scheduled Learning And Teaching ActivitiesLecture201:0020:0010 Formal Lectures – On-Line; 10 Formal Lectures – Present in Person;
Scheduled Learning And Teaching ActivitiesLecture21:002:001 Revision Lecture – On-line; 1 Revision Lecture – Present in Person
Scheduled Learning And Teaching ActivitiesLecture51:005:00Problem Classes – Present in Person
Guided Independent StudyAssessment preparation and completion151:0015:00Completion of in course assessments
Scheduled Learning And Teaching ActivitiesDrop-in/surgery51:005:00Synchronous On-Line
Guided Independent StudyIndependent study531:0053:00Preparation time for lectures, background reading, coursework review
Total100:00
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 Examination1201A80N/A
Other Assessment
Description Semester When Set Percentage Comment
Computer assessment1M10CBA - NUMBAS
Prob solv exercises1M10Problem 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