MA2CANU-Complex Analysis

Module Provider: Mathematics and Statistics
Number of credits: 10 [5 ECTS credits]
Level:5
Semesters in which taught: Semester 2 module
Pre-requisites: MA0FMNU Foundations of Mathematics and MA0MANU Mathematical Analysis and MA1RA1NU Real Analysis 1 and MA1RA2NU Real Analysis II and MA1LANU Linear Algebra
Non-modular pre-requisites:
Co-requisites:
Modules excluded:
Current from: 2022/3

Module Convenor: Dr Jochen Broecker
Email: j.broecker@reading.ac.uk

Type of module:

Summary module description:

This module provides an introduction to complex analysis. The students will learn something new and see how the applications of complex analysis reinforce the key ideas. The module moves gradually and steadily from complex numbers to complex-valued functions to integrals and series. The principles students learn in the class could be helpful for them in subsequent courses.Ìý

The Module lead at NUIST is Dr Jian DingÌýdf2001101@126.com).

Aims:

To introduce students to complex analysis and enable them to use complex variable techniques, particularly in some cases where the original problem does not involve complex numbers.

Assessable learning outcomes:

By the end of the module, students are expected to be able to:

• solve problems involving holomorphic functions;


• recognise and be able to apply the complex exponential and logarithm;


• evaluate path integrals of complex functions;


• identify singularities and residues of holomorphic functions;


• calculate appropriate real integrals using complex techniques.

Additional outcomes:

By the end of the module the student will begin to understand and recognise some of the structure of holomorphic functions.

Outline content:

Differentiable functions of a complex variable are remarkably well-behaved, and most of the technical complications of the real case do not arise with complex functions. This leads to some remarkably powerful results, and it turns out that complex variable techniques often offer the simplest method of evaluating certain real integrals. The notion of complex differentiability relates closely with power series. Contour integration in the complex plane will be introduced and the remarkable theorem of Cauchy established, from which a whole range of applications follow. Some applications to the evaluation of real integrals are given.

Brief description of teaching and learning methods:

Lectures supported by problem sheets and lecture-based tutorials.