MA24412 Analysis III
Introductory description
This covers three topics: (1) Riemann integration, (2) convergence of sequences and series of functions, (3) introduction to complex valued functions.
The idea behind integration is to compute the area under a curve. The fundamental theorem of calculus gives the precise relation between integration and differentiation. However, integration involves taking a limit, and the deeper properties of integration require a precise and careful analysis of this limiting process. This module proves that every continuous function can be integrated, and proves the fundamental theorem of calculus. It also discusses how integration can be applied to define some of the basic functions of analysis and to establish their fundamental properties.
Many functions can be written as limits of sequences of simpler functions (or as sums of series): thus a power series is a limit of polynomials, and a Fourier series is the sum of a trigonometric series with coefficients given by certain integrals. The second part of the module develops methods for deciding when a function defined as the limit of a sequence of other functions is continuous, differentiable, integrable, and for differentiating and integrating this limit. Norms are used at several stages and finally applied to show that a Differential Equation has a solution.
The final part of module focuses on complex valued functions, starting with the notion of complex differentiability. The module extends the results from Analysis II on power series to the complex case. The final section focuses on contour integrals, where a complex valued function is integrated along a curve. Cauchy's integral formula will be developped and a series of applications presented (to compute integrals of real valued functions, Liouville's Theorem and the Fundamental Theorem of Algebra).
Module aims
To develop a good working knowledge of the construction of the Riemann integral in one variable;
To study the continuity, differentiability and integral of the limit of a uniformly convergent sequence of functions;
To extend the results from Analysis II on power series to the complex case.
To develop the notion of contour integration for complex valued functions, and explore several applications (Cauchy's integral formula, Liouville's Theorem, the fundamental theorem of algebra,...)
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
 Riemann Integration
 Definition of the Riemann integral
Fundamental properties of the Riemann integral  The Fundamental Theorem of Calculus
 Improper integrals
 The Cantor Set and the devil’s staircase
 Definition of the Riemann integral
 Sequences and Series of Functions
 Pointwise and uniform convergence
 Series of functions
 A continuous, nowhere differentiable function
 Space filling curves
 Absolute Continuity
 Complex Analysis
 Review of basic facts about C
 Power Series
 The exponential and the circular functions
 Argument and Log
 Complex integration, contour integrals
 Links with MA259
 Consequences of Cauchy’s Theorem
 Applications of Cauchy’s formula to evaluate integrals in R
Learning outcomes
By the end of the module, students should be able to:
 To develop a good working knowledge of the construction of the Riemann integral;
 To understand the fundamental properties of the integral; main ones include: any continuous function can be integrated on a bounded interval and the Fundamental Theorem of Calculus (and its applications);
 To understand uniform and pointwise convergence of functions together with properties of the limit function;
 To study the continuity, differentiability and integral of the limit of a uniformly convergent sequence of functions;
 To study complex differentiability (CauchyRiemann equations) and complex power series;
 To study contour integrals: Cauchy's integral formulas and applications.
Indicative reading list
 Lecture notes will be provided for the module.
 The module webpage contains additional references that students may consult.
Students registered for this module may access the relevant chapters of books scanned under copyright.
Subject specific skills
 Working knowledge of the theory of Riemann integration.
 Theory for series and sequences, including the development of the notions of convergence and uniform converge for sequences and series of functions
 Working knowledge of complex analysis, to include power series, exponential and circular maps, contour integration.
 Applications of Cauchy's formula to compute integrals in R.
Transferable skills
 The students will be able to apply abstract notions in a variety of different contexts.
 Use a variety of techniques to compute complicated integrals or asymptotic expansions for functions/quantities arising from a wide range of applications in the physical sciences.
 Students will develop an ability to analyse and process complex information, triaging key concepts and effectively prepare plans for solving problems.
Study time
Type  Required 

Lectures  30 sessions of 1 hour (25%) 
Tutorials  9 sessions of 1 hour (8%) 
Private study  81 hours (68%) 
Total  120 hours 
Private study description
81 hours private study, revision for exams, and assignments
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Assessment group D2
Weighting  Study time  

Assignments  15%  
Inperson Examination  85%  

Assessment group R
Weighting  Study time  

Inperson Examination  Resit  100%  

Feedback on assessment
Marked assignments and exam feedback.
Courses
This module is Core for:
 Year 2 of UMAAG105 Undergraduate Master of Mathematics (with Intercalated Year)

UMAAG100 Undergraduate Mathematics (BSc)
 Year 2 of G100 Mathematics
 Year 2 of G100 Mathematics
 Year 2 of G100 Mathematics

UMAAG103 Undergraduate Mathematics (MMath)
 Year 2 of G100 Mathematics
 Year 2 of G103 Mathematics (MMath)
 Year 2 of G103 Mathematics (MMath)
 Year 2 of UMAAG106 Undergraduate Mathematics (MMath) with Study in Europe
 Year 2 of UMAAG1NC Undergraduate Mathematics and Business Studies
 Year 2 of UMAAG1N2 Undergraduate Mathematics and Business Studies (with Intercalated Year)
 Year 2 of UMAAGL11 Undergraduate Mathematics and Economics
 Year 2 of UECAGL12 Undergraduate Mathematics and Economics (with Intercalated Year)

UMAAGV17 Undergraduate Mathematics and Philosophy
 Year 2 of GV17 Mathematics and Philosophy
 Year 2 of GV17 Mathematics and Philosophy
 Year 2 of GV17 Mathematics and Philosophy
 Year 2 of UMAAGV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations

UPXAGF13 Undergraduate Mathematics and Physics (BSc)
 Year 2 of GF13 Mathematics and Physics
 Year 2 of GF13 Mathematics and Physics

UPXAFG31 Undergraduate Mathematics and Physics (MMathPhys)
 Year 2 of FG31 Mathematics and Physics (MMathPhys)
 Year 2 of FG31 Mathematics and Physics (MMathPhys)
 Year 2 of UMAAG101 Undergraduate Mathematics with Intercalated Year
This module is Core optional for:

UMAAGV17 Undergraduate Mathematics and Philosophy
 Year 2 of GV17 Mathematics and Philosophy
 Year 2 of GV17 Mathematics and Philosophy
 Year 2 of GV17 Mathematics and Philosophy

UMAAGV18 Undergraduate Mathematics and Philosophy with Intercalated Year
 Year 2 of GV18 Mathematics and Philosophy with Intercalated Year
 Year 2 of GV18 Mathematics and Philosophy with Intercalated Year
This module is Core option list C for:
 Year 2 of UMAAGV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations
This module is Option list B for:
 Year 4 of USTAY603 Undergraduate Mathematics,Operational Research,Statistics,Economics (with Intercalated Year)