MA2034A ANALYSIS

Module Code: MA2034A
Module Title: Analysis
Semester: Autumn
Level: 2
Lecturer: Dr. M. K. Warby
Credit value: 10
Prerequisites: MA1005A (Calculus 1), MA1006A (Calculus 2), MA1008S (Calculus 4) and MA1051S (Numerical Methods) are helpful.
Method of delivery: 24 lectures, 10 seminars and 2 revision classes.

There are 3 contact hours each week with 3 lectures in each of weeks 1 and 2. In subsequent weeks there are 2 lectures and 1 seminar per week with the revision classes in the last week of the teaching period.

Formative feedback: Participation in the seminars.
Method of assessment: Two hour unseen written examination involving a choice of three questions from five (80%) and 2 assignments (20%).

AIMS:

OBJECTIVES:

On successful completion of the module, students will have the following:

MODULE CONTENT:

REVISION OF THE NOTATION:
Notation for the natural numbers, integers, rationals and real numbers. Set notation including the union and intersection symbols.

SOME PROPERTIES OF THE REAL NUMBERS:
Supremum and infimum. Open and closed intervals. The nested interval property.

SEQUENCES OF REAL NUMBERS:
\epsilon--N definition of a limit. Combining convergent sequences. Standard convergent sequences such as (r^n) when |r|<1 and (1/n^p) when p>0. Bounded sequences. Monotonic sequences and the Monotone Convergence Theorem. Subsequences and the Bolzano Weierstrass Theorem. Cauchy sequences and the Cauchy criterion for convergence.

CONTINUOUS FUNCTIONS OF A REAL VARIABLE:
The \epsilon--\delta definition of continuity at a point. The sequential definition of continuity at a point. Lipschitz functions and differentiable functions.

THE CONTRACTION MAPPING THEOREM FOR FUNCTIONS OF A REAL VARIABLE.

SERIES OF REAL NUMBERS.
The geometric series and harmonic series. The integral test. The comparison test. The ratio test and the root test. Absolute and conditional convergence.

SEQUENCES AND SERIES OF FUNCTIONS:
The uniform norm. Pointwise and uniform convergence of a sequence of functions. Cauchy's criterion for uniform convergence. Uniform convergence preserves continuity. Series of functions and the Weierstrass M-test. Power series including the radius of convergence and Taylor's series.

LECTURE NOTES

A paper version of notes, exercises and solutions to exercises are distributed during the lectures and seminars. These are also available on the www using the URL
http://www.brunel.ac.uk/~icstmkw/2034/

READING LIST

Core Texts:

None but several texts cover most parts of the module. Several of the texts cover the material in a more rigorous way than will be covered in this module.

Supplementary Texts:

Teaching Schedule

Lecture number (approx) Topics
1--2 Introduction and overview of the module.
2 REVISION OF THE NOTATION:
Notation for the natural numbers, integers, rationals and real numbers. Set notation including the union and intersection symbols.
3 SOME PROPERTIES OF THE REAL NUMBERS:
Supremum and infimum. Open and closed intervals. The nested interval property.
4--8 SEQUENCES OF REAL NUMBERS:
\epsilon--N definition of a limit. Combining convergent sequences. Standard convergent sequences such as (r^n) when |r|<1 and (1/n^p) when p>0. Bounded sequences. Monotonic sequences and the Monotone Convergence Theorem. Subsequences and the Bolzano Weierstrass Theorem. Cauchy sequences and the Cauchy criterion for convergence.
9--11 CONTINUOUS FUNCTIONS OF A REAL VARIABLE:
The \epsilon--\delta definition of continuity at a point. The sequential definition of continuity at a point. Lipschitz functions and differentiable functions.
12--13 THE CONTRACTION MAPPING THEOREM FOR FUNCTIONS OF A REAL VARIABLE.
14--16 SERIES OF REAL NUMBERS.
The geometric series and harmonic series. The integral test. The comparison test. The ratio test and the root test. Absolute and conditional convergence.
17--22 SEQUENCES AND SERIES OF FUNCTIONS:
The uniform norm. Pointwise and uniform convergence of a sequence of functions. Cauchy's criterion for uniform convergence. Uniform convergence preserves continuity. Series of functions and the Weierstrass M-test. Power series including the radius of convergence and Taylor's series.
23--26 Revision of the module.