Complex Analysis

 Unit code: MATH20142 Credit Rating: 10 Unit level: Level 2 Teaching period(s): Semester 2 Offered by School of Mathematics Available as a free choice unit?: N

Prerequisite

Aims

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

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'.

Learning outcomes

On completion of this unit successful students will be able to:

• define continuity and differentiability for complex functions,
• prove the Cauchy-Riemann equations and apply them to complex functions in order to determine whether a given continuous function is complex differentiable,
• compute the radius of convergence for complex power series,
• evaluate integrals along a path - directly from the definition and also via the Fundamental Theorem of Contour Integration and Cauchy's Theorem,
• compute the Taylor and Laurent expansions of simple functions, determining the nature of the singularities and calculating residues,
• prove the Cauchy Residue Theorem and use it to evaluate integrals.

Assessment methods

• Other - 20%
• Written exam - 80%

Assessment Further Information

• Coursework; Weighting within unit 20%
• 2 hours end of semester examination; Weighting within unit 80%

Syllabus

1.Series. Complex series, power series and the radius of convergence. [2 lectures]

2.Continuity. Continuity of complex functions [2]

3.The complex plane. The topology of the complex plane, open sets, paths and continuous functions. [2]

4.Differentiation. Differentiable complex functions and the Cauchy-Riemann equations. [2]

5.Integration. Integration along paths, the Fundamental Theorem of Calculus, the Estimation Lemma, statement of Cauchy's Theorem. [4]

6.Argument and Logarithm. [2]

7.Taylor and Laurent Series. Cauchy's Integral Formula and Taylor Series, Zeros and Poles, Laurent Series. [3]

8.Residues. Cauchy's Residue Theorem, the evaluation of definite integrals and summation of series. [5]

Ian Stewart and David Tall, Complex Analysis, Cambridge University Press, 1983.

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.

Study hours

• Lectures - 22 hours
• Tutorials - 11 hours
• Independent study hours - 67 hours

Teaching staff

Nikita Sidorov - Unit coordinator