# KL5004 - Complex Variables

## What will I learn on this module?

The module is designed for you to develop your understanding of the principles, techniques and applications of complex variables.

The module will cover topics such as:

Complex numbers: Basic algebraic properties and operations, Trigonometric and exponential forms; Products and powers; Stereographic projection.

Functions of one complex variable: Limits, continuity and mappings; Differentiability, analyticity of a function and Cauchy-Riemann equations; Exponential and trigonometric functions; branches and derivatives of logarithms.

Integrals: Contours, contour integrals, Cauchy’s integral theorem and formula; Liouville’s theorem; Fundamental theorem of algebra.

Series: Convergence of sequences and series; Laurent series; Integration and differentiation of power series.

Residues and poles: Types of isolated singular points; Cauchy’s residue theorem; Meromorphic functions; Applications of residues.

### How will I learn on this module?

The module will be delivered using a combination of lectures and problem classes. A set of lecture notes and essential readings will be provided to support your learning. Problem classes will complement each taught topic and these sessions will enable you to enhance your problem-solving skills and obtain direct guidance and feedback.

You will be assessed by a written mid-term assignment (30%) and a formal examination (70%) at the end of the module. The latter will cover all aspects of the module and will assess your problem-solving abilities based on the use of a wide spectrum of complex calculus techniques.

Informal feedback on your work will be given continuously during the module. You will also receive formal assignment feedback at the end of the first semester and examination feedback at the end of the module.

### How will I be supported academically on this module?

Lectures and problem-solving classes will be the main point of academic contact, providing you with a formal teaching environment for core learning. Outside formal scheduled teaching, you will be able to contact the module team (module tutor, year tutor, programme leader) either via email or the open-door policy operated throughout the programme. Further academic support will be provided through technology-enhanced resources via the e-learning portal. You will also have the opportunity to give your feedback formally through periodic staff-student committees and directly to the module tutor at the end of the lecture.

### What will I be expected to read on this module?

All modules at Northumbria include a range of reading materials that students are expected to engage with. The reading list for this module can be found at: http://readinglists.northumbria.ac.uk
(Reading List service online guide for academic staff this containing contact details for the Reading List team – http://library.northumbria.ac.uk/readinglists)

### What will I be expected to achieve?

Knowledge & Understanding:
1. Perform algebraic manipulations with complex numbers and functions of complex variable. .
2. Evaluate basic contour integrals in the complex plane. Use Cauchy’s residue theorem to evaluate certain types of definite and improper integrals occurring in real analysis and applied mathematics

Intellectual / Professional skills & abilities:

3. Construct rigorous mathematical arguments to produce relatively complex calculations, understanding their effectiveness and range of applicability
4. Discuss and critically evaluate analytical approaches to formulating challenging problems of real analysis using methods of Complex Analysis

Personal Values Attributes (Global / Cultural awareness, Ethics, Curiosity) (PVA):
5. Demonstrate the ability to manage time and resources effectively

### How will I be assessed?

SUMMATIVE
1. Assignment (30%) – 1, 4, 5
Examination (70%) – 2, 3, 4

Summative assessment consist of coursework - to assess independent learning and analytical skills – and formal examination with a focus on problem solving skills.

FORMATIVE
problem-solving classes – 1, 2, 3, 4, 5

Feedback will take several forms, including verbal feedback after lectures, individual verbal and written feedback following written assignment, and written feedback on the exam.

None

None

### Module abstract

Calculus in the complex plane is one of the most fundamental branches of mathematics with a wide range of applications in mathematical modelling, physics and engineering. The module will enable you to formulate mathematical problems in the language of functions of a complex variable using a combination of lectures and seminars. Lectures will focus on providing you with a thorough presentation of the theory and analytical tools underlying complex numbers and functions. You will then build upon these concepts in practical sessions to acquire advanced problem-solving skills through a variety of research-informed questions that will deepen your understanding of fundamental notions as well as highlight significant applications. You will receive constructive feedback during lectures and problem solving sessions throughout the year. The assessment will consist in an application-oriented assignment and a final examination that are designed to put into light your newly developed skills and techniques. The concepts and practical abilities gained in this module will advance your fundamental knowledge of mathematics and enhance your postgraduate perspectives.

### Course info

UCAS Code G100

Credits 20

Level of Study Undergraduate

Mode of Study 3 years Full Time or 4 years with a placement (sandwich)/study abroad

Department Mathematics, Physics and Electrical Engineering

Location City Campus, Northumbria University

City Newcastle

Start September 2024 or September 2025

## Mathematics BSc (Hons)

All information is accurate at the time of sharing.

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