Analysis - 2019 entry

Infinite processes appear naturally in many contexts, from science and engineering to economics. From solving the equation that finds the wave function of a quantum system in physics, processing sensor data in engineering, to calculating prices for options in economics, at the foundation of all of these are infinite processes and the pure mathematics developed to rigorously and correctly handle these processes.  That field of pure mathematics is analysis, and the central object of study in analysis is the limit which further extends to the notions of convergence, continuity, differentiation, and integration.  

In this module, building on material from the first year, we will carefully and rigorously develop notions first in the context of real variables and then in the context of complex variables.  In particular, we will develop and you will learn how to rigorously handle real-variable differentiation, Riemann integration, analyticity, contour integration, power series, and the basic notions of topology. Quite surprisingly, in many ways complex analysis turns out to be more elegant than real analysis!

The material in this module will provide a foundation for the study of geometry/topology, number theory, dynamical systems, differential equations, probability, harmonic analysis, etc., in pure mathematics as well as the basis for applications in economics, science, and engineering. 

Pre-requisite modules: MTH1001; MTH1002

Analysis is the theory that underpins all continuous mathematics. The objective of this module is to provide you with a logically based introduction to real and 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 analysis using a rigorous approach;

2 develop proofs related to basic topological concepts like connectedness;

3 compare and contrast the theory of analytic functions over the real and complex numbers;

4 compute contour integrals and to apply this to real analysis;

Discipline Specific Skills and Knowledge:

5 apply fundamental mathematical concepts, manipulations and results in analysis;

6 formulate rigorous arguments as part of your mathematical development;

Personal and Key Transferable/ Employment Skills and Knowledge:

7 think analytically and use logical argument and deduction;

8 communicate your ideas effectively in writing and verbally;

9 manage your time and resources effectively.

- Review of epsilon-N sequence limits; monotone convergence; Bolzano-Weierstrass; Cauchy sequences and completeness;

- Epsilon-delta function limits; continuity; differentiability (all in R then R^n);

- Function classes: C^k, C^infinity etc; Lipschitz continuity;

- Complex analysis; Cauchy-Riemann equations; contrast to real analytic functions;

- Inverse and implicit function theorems;

- Formal theory of Riemann integration; integrability of monotonic functions and continuous functions; problems interchanging limits in general; pointer towards Lebesgue integration and monotone/dominated convergence theorems;

- 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;

- Rouche's theorem, principle of the argument;

- Applications to definite integrals, summation of series and the location of zeros.

In the case of module referral, the higher of the original assessment and the reassessment will be recorded for each component mark. In the case of module referral, the final mark for the module reassessment will be capped at 40%.

Basic reading:

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

Web based and Electronic Resources:

William F. Trench, Introduction to Real Analysis, freely downloadable here: https://digitalcommons.trinity.edu/mono/7/

Other Resources:

Reading list for this module: