MA4CAN: Complex Analysis

Module code: MA4CAN

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

Credits: 20

Level: 7

When you’ll be taught: Semester 2

Module convenor: Dr Jochen Broecker, email: j.broecker@reading.ac.uk

Pre-requisite module(s): BEFORE TAKING THIS MODULE YOU MUST TAKE MA1RA1 AND TAKE MA1FM AND TAKE MA2RAT (Compulsory)

Co-requisite module(s):

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

Module(s) excluded: IN TAKING THIS MODULE YOU CANNOT TAKE MA3CAN OR TAKE MA3CA2 (Compulsory)

Placement information: NA

Academic year: 2026/7

Available to visiting students: Yes

Talis reading list: Yes

Last updated: 13 April 2026

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 in-teresting 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 students are expected to be able to:

1. demonstrate self-direction and originality in tackling and solving problems involving holomorphic func-tions and power series on the complex plane
2. evaluate path integrals of complex functions and identify singularities and residues of holomorphic functions
3. demonstrate originality in the application of techniques from complex analysis in other fields of mathe-matics and mathematical physics, including the evaluation of real integrals using complex techniques
4. learn independently as is required for continuous personal development in the area of complex analy-sis and its applications throughout other branches of mathematics.

Module content

• Revise definitions and basic properties of complex numbers.
• Basic topology on the complex plane.
• Basic properties of manifolds.
• 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.
• 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 50% 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 a DAS-agreed extension as a reasonable adjustment indicated in your Individual Learning Plan): 10% of the total marks available for that piece of work will be deducted from the mark for each calendar day (or part thereof) following the deadline up to a total of three calendar days;
• where the piece of work is submitted up to three calendar days after the original deadline (or a DAS-agreed extension as a reasonable adjustment indicated in you Individual Learning Plan), 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 calendar days after the original deadline (or a DAS-agreed extension as a reasonable adjustment indicated in your Individual Learning Plan), no penalty shall be imposed;
• where the piece of work is submitted more than three calendar days after the original deadline (or a DAS-agreed extension as a reasonable adjustment indicated in your Individual Learning Plan): a mark of zero will be recorded.

Assessments marked Pass/Fail

• where the piece of work is submitted within three calendar days of the deadline (or a DAS-agreed extension as a reasonable adjustment indicated in your Individual Learning Plan): no penalty will be applied;
• where the piece of work is submitted more than three calendar days after the original deadline (or a DAS-agreed extension as a reasonable adjustment indicated in your Individual Learning Plan): a grade of Fail will be awarded.

Where a piece of work is submitted late after a deadline which has been revised owing to an extension granted through the Assessment Adjustments policy and process (self-certified or otherwise), it will be subject to the maximum penalty (i.e., considered to be more than three calendar days late). This will also apply when such an extension is used in conjunction with a DAS-agreed extension as a reasonable adjustment.

The University policy statement on penalties for late submission can be found at: /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