Complex Variables and Applications

The details
Mathematical Sciences
Colchester Campus
Undergraduate: Level 6
Thursday 08 October 2020
Friday 18 December 2020
15 July 2020


Requisites for this module



Key module for

BSC G100 Mathematics,
BSC G102 Mathematics (Including Year Abroad),
BSC G103 Mathematics (Including Placement Year),
BSC G104 Mathematics (Including Foundation Year),
MMATG198 Mathematics,
BSC G1G4 Mathematics with Computing (Including Year Abroad),
BSC G1G8 Mathematics with Computing (Including Foundation Year),
BSC G1GK Mathematics with Computing,
BSC G1IK Mathematics with Computing (Including Placement Year),
BSC G1F3 Mathematics with Physics,
BSC G1F4 Mathematics with Physics (Including Placement Year),
BSC GCF3 Mathematics with Physics (Including Year Abroad)

Module description

An introduction to complex analysis, up to and including evaluation of contour integrals using the Residue theorem.

Module aims

To introduce functions of a complex variable and techniques for complex integration including Cauchy’s theorem, integral formula, residue formula, and Jordan’s Lemma.

Module learning outcomes

On successful completion of the course, students should be able to:

- express complex numbers in both Cartesian and polar forms;
- identify curves and regions in the complex plane defined by simple formulae;
- determine whether and where a function is holomorphic / analytic;
- carry our complex integration via line integrals, Cauchy’s Theorem, Cauchy’s integral formula and Cauchy’s differentiation formula.
- obtain appropriate series expansions of functions;
- evaluate residues at pole singularities;
- apply the Residue Theorem to the calculation of real integrals.

Module information


Complex numbers:
- Cartesian and polar forms
- Lines, circles and regions in the complex plane

Functions of a complex variable:
- derivatives
- holomorphic functions
- Cauchy-Riemann Equations

Complex Integration:
- Line integrals
- Cauchy's theorem
- Cauchy's integral formula
- Derivatives of an analytic function (Cauchy's differentiation formula)

Sequences and Series of Complex Numbers:
- Taylor series
- Analytic functions and their relationship to holomorphic functions
- Laurent's theorem

Residue Integration Methods:
- Calculation of residues at poles
- Cauchy's residue theorem
- Jordan's lemma
- Calculation of definite integrals using residue theory.

Learning and teaching methods

Teaching will be delivered in a way that blends face-to-face classes, for those students that can be present on campus, with a range of online lectures, teaching, learning and collaborative support.


This module does not appear to have any essential texts. To see non-essential items, please refer to the module's reading list.

Assessment items, weightings and deadlines

Coursework / exam Description Deadline Weighting
Coursework   Assignment 1     
Coursework   Assignment 2     
Exam  120 minutes during Summer (Main Period) (Main) 

Overall assessment

Coursework Exam
20% 80%


Coursework Exam
20% 80%
Module supervisor and teaching staff
Prof Christopher Saker, email:
Professor Chris Saker & Dr David Penman
Professor Christopher Saker (, Dr David Penman (



External examiner

No external examiner information available for this module.
Available via Moodle
No lecture recording information available for this module.


Further information
Mathematical Sciences

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