Last updated 6th February 2013
The other half of this course, on Complex Analysis, will be given by Dr C. Walkden.
The official source of material for this course is the syllabus page, but here I will give more details of the course and its examination procedures. This page is a place of resources for students on the course.
Schuster Bragg Th
Roscoe Th A
Crawford Th 1
Roscoe Th B
You are, of course, welcome to make an appointment by email and come an see me in my office if you should have any problems with the course.
The exam paper can be found here
In this document you will find the solutions along with a description of some of the commonly seen mistakes.
The course naturally falls into four parts of Limits, Continuity, Differentiation and Integration. After each part I will make available notes on that section. These notes will state carefully all definitions and Theorems. Hopefully the notes will also give motivation for the results. But the notes will not contain proofs of the theorems nor details of applications of the results in examples. There has to be some reason for attendance at lectures.
You need to recall some results and techniques from courses in your First Year.
For instance you will need to know the triangle inequality, |a+b| ≤ |a|+|b| for all real numbers a,b. Perhaps you can use this to show that |a-b|≥ ||a|-|b||?
For Trigonometric Functions you will not need to know more than sinθ and cosθ can be defined as ratios of lengths of sides in a right angled triangle. From such a definition it is simple to deduce –1 ≤ sinθ ≤ 1, –1 ≤ cosθ ≤ 1 and sin2θ + cos2θ = 1 for all θ, the latter just being a restatement of Archimede's Theorem that the square of the hypotenuse is equal to the sum of the squares on the other two sides.
You will need to know what is meant by the greatest lower bound and least upper bound for a set S. Then you will need know the Completeness Property of R, that a non-empty set bounded above has a least upper bound, and a non-empty set bounded below has a greatest lower bound.
For Sequences of Real Numbers you will not need to know much more than the definition of the limit of a sequence, the Sum, Product and Quotient Rules for limits along with the particular example that limn → ∞ xn/n! =0 for all real x.
For Finite Series you will need to know the formula for ∑i=1nik for k=1, 2 and 3, along with the formula for ∑i=1nxi valid for x ≠1.
For Infinite Series of real numbers a1+ a2+ a3+ ... you will need to know what is meant by saying that it converges, namely that the sequence of partial sums, s1 = a1, s2 = a1+ a2, s3 = a1+ a2+ a3 , ... , sn =∑i=1nai, ... converges.
For Power Series you will need to know what is meant by the radius of convergence. In particular you should know the power series 1+ x + x2/2! + x3/3! + ... + xn/n! +... for the exponential function, ex, and that it has an infinite radius of convergence, i.e. converges for all real x.
In Differentiation you should know the derivatives of standard functions, i.e. the exponential function ex, the hyperbolic functions, sinhx etc., the trigonometric functions, sinθ, etc., and the logarithm, lnx. The point of this course is to give justifications for what you already know for differentiation. Similarly, we will give proofs of results you should already be familiar with, namely the Product, Quotient and Chain Rules. And by "know" I mean "be able to use" and the only way you can use these results effectively is to practice on many examples.
Here are four sheets of questions, they are not for revision after the course but to help preparation before.
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|Notes Part 1.1||1.1 Limits limit of a real-valued function at a finite point, one-sided limits, limits of a function at infinity.|
|Notes Part 1.2||1.2 Limits Divergence. Limits Rules, including the Sum, Product and Quotient rules, the Sandwich Rule and the Composite rule. Special limits.|
|Notes Part 2.1||2.1 Continuous functions Continuous functions and their properties. Definition. Continuity Rules, including the Sum, Product and Quotient rules and the Composite rule.|
|Notes Part 2.2 v2||2.2 Continuous functions Properties of continuous functions, including the Intermediate Value Theorem, and the Boundedness Theorem. Monotonic functions, the Inverse Function Theorem.|
|Notes Part 3.1||3.1 Differentiation Definition. Rules for differentiation, including the Sum, Product and Quotient rules, the Chain Rule and the Inverse Rule.|
|Notes Part 3.2||3.2 Differentiation Derivative Results including: Rolle's Theorem, the Mean Value Theorem, Cauchy's Mean Value Theorem and L'Hôpital's Rule.|
|Notes Part 3.3||3.3 Differentiation Taylor Polynomials and Taylor's Theorem with Cauchy's and Lagrange's forms of the error. Taylor Series and a number of Standard series.|
|Notes Part 4.1||3.4 Integration Definition and examples. Criterion for being Riemann Integrable.|
An important difference between Real and Complex Analysis is that you may well have functions in real analysis that have n derivatives but not n+1 derivatives. Examples of such functions can be constructed using the sin(π/x) and cos(π/x) functions.
Here is a powerpoint-type presentation of the Proof of the Intermediate Value Theorem. I would be grateful of any feedback - positive or negative, though in the latter case I would prefer constructive criticism!
If you wish to print the proof I would suggest looking at
Here are a couple of examples of a function that is everywhere continuous but nowhere differentiable:
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