2P1: Complex Analysis - course unit details - BSc Computer Science and Mathematics with Industrial Experience - full details (2025 entry)

Duration: 4 years
Year of entry: 2025
UCAS course code: GG41 / Institution code: M20

Key features:
Industrial experience
Scholarships available

Typical A-level offer: A*A*A including specific subjects
Typical contextual A-level offer: AAA including specific subjects
Refugee/care-experienced offer: AAB including specific subjects
Typical International Baccalaureate offer: 38 points overall with 7,7,6 at HL, including specific requirements

Full entry requirements How to apply

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Fees and funding

Fees

Tuition fees for home students commencing their studies in September 2025 will be £9,535 per annum (subject to Parliamentary approval). Tuition fees for international students will be £36,000 per annum. For general information please see the undergraduate finance pages.

Policy on additional costs

All students should normally be able to complete their programme of study without incurring additional study costs over and above the tuition fee for that programme. Any unavoidable additional compulsory costs totalling more than 1% of the annual home undergraduate fee per annum, regardless of whether the programme in question is undergraduate or postgraduate taught, will be made clear to you at the point of application. Further information can be found in the University's Policy on additional costs incurred by students on undergraduate and postgraduate taught programmes (PDF document, 91KB).

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The University of Manchester is committed to attracting and supporting the very best students. We have a focus on nurturing talent and ability and we want to make sure that you have the opportunity to study here, regardless of your financial circumstances.

For information about scholarships and bursaries please visit our undergraduate student finance pages .

Course unit details:
2P1: Complex Analysis

Course unit fact file
Unit code	MATH29141
Credit rating	10
Unit level	Level 2
Teaching period(s)	Semester 1
Available as a free choice unit?	No

Overview

This course introduces the analysis of complex functions of a complex variable. Complex differentiability is a very strong condition, and differentiable (or holomorphic or regular) functions have many strong properties. Integration is along paths in the complex plane. The central result of this spectacularly beautiful part of mathematics is Cauchy’s Theorem guaranteeing that certain integrals along closed paths are zero. After studying properties of isolated singularities of functions, Cauchy’s theorem leads to useful techniques for evaluating real integrals based on the ‘calculus of residues’ (problems unsolvable by other means).

Syllabus:

A.   Complex functions: Domains and paths in the complex plane. Differentiation and the Cauchy-Riemann equations; holomorphic functions. Path integrals and the Fundamental Theorem of contour integration.
B.   Power series: Power series and their radius of convergence. Derivatives and integrals of power series. Elementary functions such as exp, sin, cos, sinh, cosh and log.
C.   Cauchy’s Theorem and Formula: Winding numbers, Cauchy’s theorem, Cauchy’s integral formula and the estimation lemma.
D.   Taylor and Laurent series: The Cauchy-Taylor theorem, Liouville’s theorem, Laurent’s theorem and calculation of Laurent series.
E.   Residues: Isolated singularities, poles and their residues; Cauchy’s residue theorem. Applications of the residue theorem to the evaluation of trigonometric integrals, integrals over the real line, and summation of series.

Pre/co-requisites

Unit title	Unit code	Requirement type	Description
Mathematical Foundations & Analysis	MATH11121	Pre-Requisite	Compulsory

Aims

The unit aims to introduce the basic ideas of complex analysis, with particular emphasis on contour integration, Cauchy’s Theorem and the calculus of residues.

The course is only available to students on the BSc/MMathPhys Mathematics & Physics programmes and the BSc Computer Science and Mathematics programme.

Learning outcomes

On succesful completion of the course, students will be able to:

Use the Cauchy-Riemann Theorem it to decide whether a given function is holomorphic.
Define power series, determine their properties, and use them to define elementary holomorphic functions.
Use the Cauchy-Taylor Theorem and Laurent's Theorem to expand a holomorphic function as a power series on a disc or on an annulus.
Define and calculate complex integrals using a variety of methods (in particular the Fundamental Theorem and the Cauchy Residue Theorem) and apply to the evaluation of some real integrals.

Teaching and learning methods

This course will be delivered together with the first half of MATH34011 (which is 20 credits). It will involve 3 hours of lectures plus an examples class and a tutorial each week for 6 weeks.

The course is only available to students on the BSc/MMathPhys Mathematics & Physics programmes and the BSc Computer Science and Mathematics programme.

Assessment methods

Method	Weight
Other	20%
Written exam	80%

Online coursework test (using stack)
Test mid-way through semester; Stack feedback available after test is closed
20% Weighting

Written exam
General feedback provided after exam is marked
80% Weighting

Feedback methods