BSc Computer Science and Mathematics / Course details
Year of entry: 2020
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Course unit details:
|Unit level||Level 2|
|Teaching period(s)||Semester 2|
|Offered by||Department of Mathematics|
|Available as a free choice unit?||No|
This course introduces the calculus of complex functions of a complex variable. It turns out that complex differentiability is a very strong condition and differentiable functions behave very well. 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. This striking result leads to useful techniques for evaluating real integrals based on the 'calculus of residues'.
|Unit title||Unit code||Requirement type||Description|
|Foundations of Pure Mathematics A||MATH10101||Pre-Requisite||Compulsory|
|Calculus and Vectors A||MATH10121||Pre-Requisite||Compulsory|
|Real Analysis A||MATH20101||Pre-Requisite||Compulsory|
MATH10101 Foundations of Pure Maths A or MATH10111 Foundation of Pure Maths B
MATH10121 Calculus and Vectors A or MATH10131 Calculus and Vectors B
MATH20101 Real Analysis A or MATH20111 Real Analysis B
The course unit aims to introduce the basic ideas of complex analysis, with particular emphasis on Cauchy's Theorem and the calculus of residues.
On completion of this unit successful students will be able to:
- prove the Cauchy-Riemann Theorem and its converse and use them to decide whether a given function is holomorphic;
- use power series to define a holomorphic function and calculate its radius of convergence;
- define the complex integral and use a variety of methods (the Fundamental Theorem of Contour Integration, Cauchy's Theorem, the Generalised Cauchy Theorem and the Cauchy Residue Theorem) to calculate the complex integral of a given function;
- define and perform computations with elementary holomorphic functions such as sin, cos, sinh, cosh, exp, log, and functions defined by power series;
- use Taylor's Theorem and Laurent's Theorem to expand a holomorphic function in terms of power series on a disc and Laurent series on an annulus, respectively;
- identify the location and nature of a singularity of a function and, in the case of poles, calculate the order and the residue;
- apply techniques from complex analysis to deduce results in other areas of mathematics, including proving the Fundamental Theorem of Algebra and calculating infinite real integrals, trigonometric integrals, and the summation of series.
1 Introduction and recap on complex numbers.
2 Limits and differentiation in the complex plane and the Cauchy-Riemann equations.
3 Power series and elementary analytic functions.
4 Complex integration and Cauchy’s Theorem.
5 Cauchy’s Integral Formula and Taylor’s Theorem.
6 Laurent series and singularities.
7 Cauchy’s Residue Theorem.
- Coursework; Weighting within unit 20%
- 2 hours end of semester examination; Weighting within unit 80%
Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
Ian Stewart and David Tall, Complex Analysis, Cambridge University Press, 1983.
|Scheduled activity hours|
|Independent study hours|
|Charles Walkden||Unit coordinator|