Complex Analysis - 2023 entry

The central object of study in analysis is the limit and related notions of convergence, continuity, differentiation, and integration. 

In this module, we carefully and rigorously develop an understanding of the analysis of functions of a complex variable. You will learn how to rigorously handle differentiation, integration, analyticity, contour integration, power series, and topology of the complex plane. Quite surprisingly, complex analysis is in many ways simpler than real analysis and has many practical applications.

The material in this module provides foundations for the study of Analytic Number Theory (MTHM041) and MTHM041 (Analytic Number Theory), etc. in pure mathematics as well as being the basis for many techniques for solving practical problems in economics, science, and engineering. Hence it is highly recommended to all mathematics students.

Pre-requisite modules: MTH2008 (or equivalent)

The objective of this module is to provide you with a logically based introduction to complex analysis. The primary objective is to define all the basic concepts clearly and to develop them sufficiently to provide proofs of useful theorems. This enables you to see the reason for studying analysis, and develops the subject to a stage where you can use it in a wide range of applications.
 

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:
1 state and prove key theorems in complex analysis using a rigorous approach;
2 understand properties of analytic functions over the complex numbers;
3 use contour integrals for computational and theoretical purposes;

Discipline Specific Skills and Knowledge:
4 apply fundamental mathematical concepts, manipulations and results in analysis;
5 formulate rigorous arguments as part of your mathematical development;

Personal and Key Transferable/ Employment Skills and Knowledge:
6 think analytically and use logical argument and deduction;
7 communicate your ideas effectively in writing and verbally;
8 manage your time and resources effectively.
 

- Epsilon-delta function limits; continuity; differentiability in the complex plane;
- Basic topology in the plane;
- Cauchy-Riemann equations; contrast to real analytic functions;
- Contour integrals; poles and singularities (isolated, removable, essential); residues; Cauchy's Theorem; Cauchy integral formulae; Taylor series and Laurent series;
- Maximum modulus principle, Liouville's theorem, fundamental theorem of algebra, meromorphic functions, residue theorem;
- Rouché’s theorem, principle of the argument;
- Applications to definite integrals, summation of series and location of zeros.
 

*Please refer to reassessment notes for details on deferral vs. Referral reassessment

Web based and Electronic Resources:

ELE: http://vle.exeter.ac.uk

Reading list for this module: