# FUNCTIONS OF A COMPLEX VARIABLE - 2017/8

Module code: MAT3034

Module provider

Mathematics

Module Leader

PRINSLOO AH Dr (Maths)

Number of Credits

15

ECTS Credits

7.5

Framework

FHEQ Level 6

JACs code

G100

Module cap (Maximum number of students)

N/A

Module Availability

Semester 1

Overall student workload

Independent Study Hours: 117

Lecture Hours: 33

Assessment pattern

Assessment type | Unit of assessment | Weighting |
---|---|---|

School-timetabled exam/test | IN-SEMESTER TEST (50 MINS) | 20 |

Examination | EXAM | 80 |

Alternative Assessment

N/A

Prerequisites / Co-requisites

None.

Module overview

This module provides an introduction to the theory of functions of a complex variable, known as Complex Analysis, which is a fundamental topic in modern mathematics and is widely used in many branches of mathematics and physics.

Module aims

The aim of this module is to introduce students to the theory of complex functions of a complex variable – including key concepts such as complex differentiation, integration along contours on the complex plane, series expansions of complex functions, and the calculus of residues. At the end of the module, students should have gained a thorough understanding of the theory of functions of a complex variable and should be able to apply this knowledge in a variety of contexts.

Learning outcomes

Attributes Developed | |
---|---|

Understand what is meant by the continuity and differentiability of complex functions of a complex variable | KCT |

Quote, derive and apply the Cauchy-Riemann equations | KCT |

Compute contour integrals of continuous complex functions | KCT |

Quote and apply Cauchy's theorem | KCT |

Quote, apply and derive Cauchy's integral formulae | KCT |

Quote, derive and apply Liouville's theorem | KCT |

Analyse the convergence properties of complex power series | KCT |

Quote, derive and apply Taylor's theorem and Laurent's theorem, and compute Taylor and Laurent series expansions of complex functions | KCT |

Identify and classify the singularities of complex functions, and compute the residues of simple and higher order poles | KCT |

Quote, derive and apply Cauchy's residue theorem, and use the calculus of residues to compute real integrals | KCT |

Attributes Developed

**C** - Cognitive/analytical

**K** - Subject knowledge

**T** - Transferable skills

**P** - Professional/Practical skills

Module content

Indicative content includes:

complex functions and complex differentiation;

the Cauchy-Riemann equations and harmonic functions;

direct contour integration;

Cauchy's theorem and integral formulae, and their applications;

Liouville’s theorem and the fundamental theorem of algebra;

Taylor and Laurent series expansions;

the classification of the singularities of complex functions;

Cauchy's residue theorem and its applications.

Methods of Teaching / Learning

The learning and teaching strategy is designed to introduce students to the theory of complex functions of a complex variable, and its applications.

The learning and teaching methods include:

3 hours of lectures per week x 11 weeks – material supplementing the course notes will be presented on a blackboard/whiteboard, with Q & A opportunities for students.

Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate

an understanding of and the ability to interpret and manipulate mathematical statements;

subject knowledge through the recall of key definitions, theorems and their proofs;

analytical ability through the solution of unseen problems in the in-semester test and exam.

Thus, the summative assessment for this module consists of:

one two hour examination at the end of Semester 2 – worth 80% module mark;

one in-semester test – worth 20% module mark.

Formative assessment and feedback

Students will receive written feedback in the form of two marked pieces of unassessed coursework. Additional verbal feedback will be provided by the lecturer at optional weekly office-hours or help-sessions.

Reading list

Reading list for FUNCTIONS OF A COMPLEX VARIABLE : http://aspire.surrey.ac.uk/modules/mat3034

Programmes this module appears in

Programme | Semester | Classification | Qualifying conditions |
---|---|---|---|

Mathematics and Physics MPhys | 1 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Economics and Mathematics BSc (Hons) | 1 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Financial Mathematics BSc (Hons) | 1 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Mathematics with Statistics BSc (Hons) | 1 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Mathematics BSc (Hons) | 1 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Mathematics MMath | 1 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Mathematics and Physics BSc (Hons) | 1 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Mathematics and Physics MMath | 1 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2017/8 academic year.