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# BSc Computer Science and Mathematics / Course details

Year of entry: 2020

## Course unit details:Complex Analysis

Unit code MATH20142 10 Level 2 Semester 2 Department of Mathematics No

### Overview

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'.

### Pre/co-requisites

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

Pre-requisites:

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

### Aims

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

### Learning outcomes

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.

### Syllabus

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.

### Assessment methods

Method Weight
Other 20%
Written exam 80%
• Coursework; Weighting within unit 20%
• 2 hours end of semester examination; Weighting within unit 80%

### Feedback methods

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.

### Study hours

Scheduled activity hours
Lectures 22
Tutorials 11
Independent study hours
Independent study 67

### Teaching staff

Staff member Role
Charles Walkden Unit coordinator