MA3CANNU: Complex Analysis

Module code: MA3CANNU

Module provider: Mathematics and Statistics; School of Mathematical, Physical and Computational Sciences

Credits: 20

Level: Level 3 (Honours)

When you'll be taught: Semester 2

Module convenor: Professor Michael Levitin, email: m.levitin@reading.ac.uk

Module co-convenor: Dr Sugata Mondal, email: s.mondal@reading.ac.uk

NUIST module lead: Jian Ding, email: df2001101@126.com

Pre-requisite module(s): BEFORE TAKING THIS MODULE YOU MUST TAKE MA0FMNU AND TAKE MA1RA1NU AND TAKE MA1RA2NU (Compulsory)

Co-requisite module(s):

Pre-requisite or Co-requisite module(s):

Module(s) excluded:

Placement information: NA

Academic year: 2024/5

Available to visiting students: No

Talis reading list: No

Last updated: 12 September 2024

Overview

Module aims and purpose

To introduce students to complex analysis, which is the theory of differentiable functions on the complex plane and with complex values (so-called holomorphic functions). These functions have a number of interesting properties which are not readily expected from Real Analysis. Most importantly, such functions can be represented with power series, and a number of important functions (such as the exponential function, trigonometric functions, and hyperbolic functions) arise in this way.

Module learning outcomes

By the end of the module, it is expected that students will be able to:

1. Solve problems involving holomorphic functions and power series on the complex plane
2. Evaluate path integrals of complex functions
3. Identify singularities and residues of holomorphic functions
4. Apply techniques from complex analysis in other fields of mathematics and mathematical physics, including the evaluation of real integrals using complex techniques

Module content

• Revise definitions and basic properties of complex numbers
• Basic topology on the complex plane
• Convergence of sequences and series
• Functions on the complex plane, continuity
• Holomorphic functions, basic properties
• Power series
• Path integration, Cauchy’s theorem, Cauchy’s integral formula
• Laurent’s theorem and isolated singularities
• Residue calculus
• Liouville’s theorem
• Schwarz’ reflection principle
• Maximum principle
• Schwarz’ lemma
• Casorati Weierstrass
• Analytical continuation
• Application to fluid dynamics, electrodynamics, and signal processing

Structure

Teaching and learning methods

The material is delivered via lectures supported by tutorials with formative exercises.

Study hours

At least 55 hours of scheduled teaching and learning activities will be delivered in person, with the remaining hours for scheduled and self-scheduled teaching and learning activities delivered either in person or online. You will receive further details about how these hours will be delivered before the start of the module.

Please note the independent study hours above are notional numbers of hours; each student will approach studying in different ways. We would advise you to reflect on your learning and the number of hours you are allocating to these tasks.

Semester 1 The hours in this column may include hours during the Christmas holiday period.

Semester 2 The hours in this column may include hours during the Easter holiday period.

Summer The hours in this column will take place during the summer holidays and may be at the start and/or end of the module.

Assessment

Requirements for a pass

Students need to achieve an overall module mark of 40% to pass this module.

Summative assessment

Penalties for late submission of summative assessment

The Support Centres will apply the following penalties for work submitted late:

Assessments with numerical marks

• where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for that piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of three working days;
• the mark awarded due to the imposition of the penalty shall not fall below the threshold pass mark, namely 40% in the case of modules at Levels 4-6 (i.e. undergraduate modules for Parts 1-3) and 50% in the case of Level 7 modules offered as part of an Integrated Masters or taught postgraduate degree programme;
• where the piece of work is awarded a mark below the threshold pass mark prior to any penalty being imposed, and is submitted up to three working days after the original deadline (or any formally agreed extension to the deadline), no penalty shall be imposed;
• where the piece of work is submitted more than three working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

Assessments marked Pass/Fail

• where the piece of work is submitted within three working days of the deadline (or any formally agreed extension of the deadline): no penalty will be applied;
• where the piece of work is submitted more than three working days after the original deadline (or any formally agreed extension of the deadline): a grade of Fail will be awarded.

The University policy statement on penalties for late submission can be found at: https://www.reading.ac.uk/cqsd/-/media/project/functions/cqsd/documents/qap/penaltiesforlatesubmission.pdf

You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Formative assessment

Formative assessment is any task or activity which creates feedback (or feedforward) for you about your learning, but which does not contribute towards your overall module mark.

Non-assessed problem sheets 

Reassessment

Additional costs