Complex Variables and Applications
Undergraduate: Level 6
Thursday 08 October 2020
Friday 18 December 2020
15 July 2020
Requisites for this module
BSC G100 Mathematics,
BSC G102 Mathematics (Including Year Abroad),
BSC G103 Mathematics (Including Placement Year),
BSC G104 Mathematics (Including Foundation Year),
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)
An introduction to complex analysis, up to and including evaluation of contour integrals using the Residue theorem.
To introduce functions of a complex variable and techniques for complex integration including Cauchy’s theorem, integral formula, residue formula, and Jordan’s Lemma.
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.
- Cartesian and polar forms
- Lines, circles and regions in the complex plane
Functions of a complex variable:
- holomorphic functions
- Cauchy-Riemann Equations
- 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.
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
||120 minutes during Summer (Main Period) (Main)
Module supervisor and teaching staff
Prof Christopher Saker, email: email@example.com.
Professor Chris Saker & Dr David Penman
Professor Christopher Saker (firstname.lastname@example.org), Dr David Penman (email@example.com)
No external examiner information available for this module.
Available via Moodle
No lecture recording information available for this module.
* Please note: due to differing publication schedules, items marked with an asterisk (*) base their information upon the previous academic year.
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