# MATU9M3 - Analysis I

SCQF Level: 9
Module Co-ordinator: Dr PS Jackson
Lecturers: Dr PS Jackson, Mr DA Smith
Availability: Autumn
Course Prerequisite: MATU9M2
SCQF Credit Value: 20 (1 module)

### Aims

To re-examine the basic concepts of calculus and give them a rigorous foundation; to provide examples which demonstrate that, while intuition is important, it can be misleading if not accompanied by precision of argument; to regularise our use of numbers; to introduce students to functions of a complex variable.

### Learning Outcomes

Students should be able to: provide rigorous justification when finding infima and suprema of sets and limits of sequences and when discussing continuity and differentiability of real functions and power series expansions; understand the need to define number systems precisely and show how this can be accomplished; manipulate complex numbers and elementary functions of a complex variable. They should also be able to: prove standard theorems in these areas; demonstrate the ability to apply theory and techniques to unseen problems without reference to notes; work independently and under a time constraint.

### Contents

#### Real analysis

• Least upper bounds and greatest lower bounds
• Sequences and the monotonic principle
• Limits of functions and continuity
• Differentiability
• The Intermediate Value Theorem, Rolle's Theorem, the Mean Value Theorem and
• L'Hôpital's Rule

#### Foundations of Mathematics

• Congruence classes, functions, rings and fields
• Construction of the natural numbers from Peano's axioms
• Construction of the integers, rationals, real numbers and complex numbers

#### Complex analysis

• Roots of complex numbers
• Polynomials, rational functions and power series
• Exponential, hyperbolic, trigonometric and logarithmic functions
• Complex powers

### Transferable Skills

Logical and analytical thinking

### Bibliography

M Hart, Guide to Analysis, Macmillan 2001
I Stewart and D Tall, The Foundations of Mathematics, OUP, 1977
S Abbott, Understanding Analysis, Springer 2001
J W Brown and R V Churchill, Complex Variables and Applications, McGraw Hill

### Teaching Format

4 one-hour lectures and a 11/2 hour tutorial per week.

### Assessment

1/3 coursework (2 class tests) and 2/3 examination.

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