# FUNCTIONS OF A COMPLEX VARIABLE - 2025/6

Module code: MAT2054

## Module Overview

This module is an introduction to the theory of complex functions of a complex variable, also known as complex analysis. We will study the continuity and differentiability of complex functions, integration along paths on the complex plane, Taylor and Laurent series expansions of complex functions with isolated singularities, and residue calculus and its applications. Complex analysis and residue calculus are widely used in many branches of Mathematics and Physics.

This module builds on material on two-variable calculus introduced in MAT1043 Multivariable Calculus.

### Module provider

Mathematics & Physics

### Module Leader

PRINSLOO Andrea (Maths & Phys)

### Number of Credits: 15

### ECTS Credits: 7.5

### Framework: FHEQ Level 5

### Module cap (Maximum number of students): N/A

## Overall student workload

Independent Learning Hours: 58

Lecture Hours: 33

Tutorial Hours: 11

Guided Learning: 15

Captured Content: 33

## Module Availability

Semester 1

## Prerequisites / Co-requisites

None.

## Module content

Indicative content includes:

- Complex functions on subsets of the complex plane.
- Differentiation of complex functions and the Cauchy-Riemann equations.
- Holomorphic complex functions.
- Branches of complex multifunctions, including complex logarithms and complex power functions.
- Paths and homotopies on the complex plane, and path-connected and simply-connected subsets.
- Contour integration.
- Cauchy's theorem and Cauchy's integral formulae, and Liouville’s theorem.
- Taylor and Laurent series expansions.
- Isolated singularities of complex functions, and residues at simple and higher order poles.
- Cauchy's residue theorem and its applications.

## Assessment pattern

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

School-timetabled exam/test | In-Semester Test (50 minutes) | 20 |

Examination | End-of-Semester Examination (2 hours) | 80 |

## Alternative Assessment

N/A

## Assessment Strategy

The __assessment strategy__ is designed to provide students with the opportunity to demonstrate:

- Understanding of subject knowledge, and recall of key definitions and results in the theory of complex analysis.
- The ability to identify and use the appropriate methods to solve problems relating to complex functions of a complex variable.

Thus, the

__summative assessment__for this module consists of:

- One in-semester test (50 minutes), worth 20% of the module mark, corresponding to Learning Outcomes 1 and 2.
- A synoptic examination (2 hours), worth 80% of the module mark, corresponding to all Learning Outcomes 1 to 6.

Formative assessment

There are two formative unassessed courseworks over an 11 week period, designed to consolidate student learning.

Feedback

Students will receive individual written feedback on both the formative unassessed courseworks and the in-semester test. The feedback is timed such that feedback from the first coursework will assist students with preparation for the in-semester test. The feedback from both courseworks and the in-semester test will assist students with preparation for the synoptic examination. Students also receive verbal feedback in tutorials and office hours.

## Module aims

- Introduce students to basic definitions and results in the theory of functions of a complex variable.
- Provide students with a thorough knowledge of holomorphic complex functions, and enable them to calculate derivatives, find isolated singularities, and determine Taylor and Laurent expansions.
- Develop students¿ understanding of contour integration and methods of solving contour integrals.
- Introduce students to residue calculus and its applications.

## Learning outcomes

Attributes Developed | ||

001 | Students will be able to demonstrate an understanding of the continuity, differentiability and holomorphicity of complex functions. | KC |

002 | Students will be able to state, derive and apply the Cauchy-Riemann equations. | KC |

003 | Students will compute contour integrals of complex functions. | KC |

004 | Students will be able to state and apply Cauchy's theorem and integral formulae, and Liouville's theorem. | KC |

005 | Students will be able to determine Taylor and Laurent expansions of complex functions. | KC |

006 | Students will be able to state and apply Cauchy's residue theorem, and use residue calculus to compute real integrals. | KCT |

Attributes Developed

**C** - Cognitive/analytical

**K** - Subject knowledge

**T** - Transferable skills

**P** - Professional/Practical skills

## Methods of Teaching / Learning

The __learning and teaching__ strategy is designed to:

- Provide students with an introduction to the theory of functions of a complex variable.
- Provide students with experience of methods used to interpret, understand and solve problems relating to complex analysis and complex functions of a complex variable.

The learning and teaching methods include:

- Three one-hour lectures for eleven weeks, with module notes provided to complement the lectures. These lectures provide a structured learning environment and opportunities for students to ask questions and to practice methods taught.
- A weekly one-hour tutorial for eleven weeks. These tutorials provide an opportunity for students to gain feedback and assistance with the exercise sheets which complement the module notes.

Two unassessed courseworks to provide students with further opportunity to consolidate learning.

- Students receive individual written feedback on these courseworks as guidance on their progress and understanding.
- Lectures may be recorded or equivalent recordings of lecture material provided. These recordings are intended to give students an opportunity to review parts of lectures which they may not fully have understood and should not be seen as an alternative to attending lectures.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

## Reading list

https://readinglists.surrey.ac.uk

Upon accessing the reading list, please search for the module using the module code: **MAT2054**

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourceness and Resilience, and Sustainability. This module is designed to allow students to develop knowledge, skills and capabilities in the following areas:

**Digital Capabilities**: The SurreyLearn page for MAT2054 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies while facilitating collaborative learning and information sharing.

**Employability**: The module MAT2054 equips students with skills which significantly enhance their employability. The mathematical proficiency gained hones critical thinking and problem-solving abilities. Students learn to evaluate complex problems, break them into manageable components, and apply logical reasoning to arrive at solutions — these are highly sought after skills in any profession.

**Global and Cultural Capabilities:** Student enrolled in MAT2054 originate from a variety of countries and have a wide range of cultural backgrounds. Students are encouraged to work together during problem-solving teaching activities in tutorials and lectures, which naturally facilitates the sharing of different cultures.

**Resourcefulness and Resilience**: MAT2054 is a module which demands a rigorous approach to Complex Analysis as well as the ability to perform lengthy calculations accurately. Students will gain skills in analysing problems and lateral thinking, and will complete assessments which challenge them and build resilience.

## Programmes this module appears in

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

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

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

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

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

Mathematics and Physics MMath | 1 | Compulsory | 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 2025/6 academic year.