Lecturer: Jose Rodrigo
Term(s): Term 1
Status for Mathematics students: Core for Maths
Commitment: 30 one-hour lectures plus three assignments
Assessment: 85% by 2-hour examination, 15% coursework
This module will be examined in the first week of Term 3
Formal registration prerequisites: None
Assumed knowledge: Notions of convergence, and basic results for sequences, series, differentiation and integration from introductory analysis modules like MA131 Analysis (or MA137 Mathematical Analysis for non-maths students); knowledge of vector spaces from MA106 Linear Algebra
Useful knowledge: Basic results about curves and surfaces and vectorfields from MA134 Geometry and Motion
Synergies: The module fits with MA259 Multivariable Calculus
Leads to: The following modules have this module listed as assumed knowledge or useful background:
- MA260 Norms, Metrics and Topologies
- MA222 Metric Spaces
- MA250 Introduction to Partial Differential Equations
- MA269 Asymptotics and Integral Transforms
- MA209 Variational Principles
- MA3H0 Numerical Analysis and PDEs
- MA3D9 Geometry of Curves and Surfaces
- MA3B8 Complex Analysis
- MA3H7 Control Theory
- MA3G1 Theory of Partial Differential Equations
- MA3K0 High Dimensional Probability
- MA3G7 Functional Analysis I
- MA359 Measure Theory
- MA4L6 Analytic Number Theory
Content: This covers three topics:
- Riemann integration
- Convergence of sequences and series of functions
- 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 developed and a series of applications presented (to compute integrals of real valued functions, Liouville's Theorem and the Fundamental Theorem of Algebra).
Objectives: By the end of the course the student should be able to:
- Develop a good working knowledge of the construction of the Riemann integral
- 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)
- Understand uniform and pointwise convergence of functions together with properties of the limit function
- Study the continuity, differentiability and integral of the limit of a uniformly convergent sequence of functions
- Study complex differentiability (Cauchy-Riemann equations) and complex power series
- Study contour integrals: Cauchy's integral formulas and applications.
There is no recommended textbook for the course. A complete set of lecture notes will be provided.