Last updated 26^{th} January 2018

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.

To give you a brief idea of what you can expect to be able to do at the end of the course here are the **Intended Learning Outcomes**:

- State the definition of limit of a function; calculate the limit for simple functions;

prove and apply the Rules for Limits to calculations for more complicated functions, - State the definition of continuity; prove that simple functions are continuous at given

points; prove and apply the Rules for Continuous functions to more complicated

functions, - State the definition of differentiable; prove that simple functions are differentiable and

calculate their derivatives at given points; prove and apply the Rules for Derivatives

to more complicated functions, - Prove and apply the Intermediate Value Theorem; Inverse function Theorem; various

results on the composition of functions; various mean value theorems, - Calculate Taylor polynomials; state Taylors Theorem with an error term; derive

bounds on the error terms; prove that Taylor series for a function converge to that

function, - State the definition of the Riemann integral; calculate the Riemann integral for various

functions.

The final **Unit Survey** response rate is 48.3%. So no new coffee mug for Charles and me (we needed 60% or more for that).

Alas, 48.3% means that 167 of the 323 students in the class did not completed the survey. But I suspect that most of the students who attended lectures completed it.

Best of luck to all in the exam, though with sufficient revision luck will not be needed. And anyway, there is no such thing as luck.

A lot of students did well on my questions, which was gratifying to see. On the attached you will find the exam solutions along with some comments on the commonly seen errors. But, for many who did not do well, the main error was not attending lectures; not learning the proofs; not attempting the problem sheets and not attending the tutorials.

For example, students had a problem with

- the limit in A1iii, though a similar limit is seen in Qu 26 on Problem Sheet 1,
- deducing the Mean Value Theorem from Rolle’s Theorem; this is Theorem 3.2.8 and the proof was given in lectures,
- the inequality in A2iii; this is Question 15 on Problem Sheet 3 and the method was discussed in lectures and a tutorial,
- the Taylor polynomial in A3, due to errors when differentiating (in turn due to lack of practice),
- remembering the definition of the Upper and Lower sums in A4; these are given in the lectures and illustrated with diagrams to help with remembering which definition has a
*lub*and which a*glb*, - the calculation of the Lower Sum in A4, due to simple errors with the algebraic manipulation of the geometric series (again due to lack of practice).

**Also**, in almost all cases where students had answered more than the required number of questions they would have been better served by searching for a mark or two extra in the correct number of questions. Especially because so many marks were lost due to arithmetic and algebraic errors which could have been caught and corrected after a careful reading.

- 2 Lectures per week. The timetable shows all the lectures concerned with the Real Analysis half of the course.
- 1 feedback tutorial per week. The timetable shows all five real analysis classes, you will have been assigned to one of these classes.

9.00 | 10.00 | 11.00 | 12.00 | 1.00 | 2.00 | 3.00 | 4.00 | |
---|---|---|---|---|---|---|---|---|

Monday | TutorialG.205 Dr. C. Dean |
|||||||

Tuesday | TutorialRoscoe 1.010 Dr. M. Coleman |
TutorialG.207 Dr. C. Dean |
TutorialSimon 2.61 Dr. M. Coleman |
|||||

Wednesday | Lecture Simon E |
|||||||

Thursday | LectureRoscoe A |
|||||||

Friday | TutorialG.207 Dr. C. Dean |

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.

**When?** The Feedback tutorials start in week 2. You will be assigned one of the five periods each week. It is important you go to your alloted tutorial, you will be recorded as not attending a class if you go to one you are not assigned.

**What happens in a tutorial?**

The Feedback tutorials will cover those Examples stated in lectures without solution.

In weeks 3, 5, 8 and 10 the tutorials will feature a Kahoot Quiz, a fun way of me seeing how well you are doing, and identifying any misconceptions the class might be having.

Otherwise, the tutorials are an opportunity for the tutor and a P.G. helper to go around the room and offer individual help. This is only of benefit if you have attempted questions **before** the tutorial.

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 sin*θ* and
cos*θ* 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 ≤ cos*θ* ≤ 1 and sin^{2}*θ* + cos^{2}θ = 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 lim_{n → ∞} x^{n}/n! =0 for all real *x*.

For *Finite Series* you will need to know the formula for
∑_{i=1}^{n}i^{k} for *k=1, 2* and *3*, along with the formula for ∑_{i=1}^{n}x^{i} valid for *x* ≠1.

For *Infinite Series* of real numbers a_{1}+ a_{2}+ a_{3}+ ... you will need to know what is meant by saying that it converges, namely that the *sequence* of partial sums, s_{1} = a_{1}, s_{2} = a_{1}+ a_{2}, s_{3} = a_{1}+ a_{2}+ a_{3} , ... , s_{n} =∑_{i=1}^{n}a_{i}, ... 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 + x^{2}/2! + x^{3}/3! + ... + x^{n}/n! +... for the exponential function, e^{x}, 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 *e ^{x}*, the hyperbolic functions, sinh

Here are four sheets of questions, they are not for revision *after* the course but to help preparation **before**.

I do not give weekly sheets. You can compare the Question Sheets with the material for that week in the lecture notes to see what questions you can attempt.

- Solution Sheet 1.1 Questions 1-11
- Solution Sheet 1.2 Questions 12-21
- Solution Sheet 1.3 Questions 22-25
- Solution Sheet 1.4 Questions 26-31

- Solution Sheet 3 Questions 1-9
- Solution Sheet 3 Questions 10-22
- Solution Sheet 3 Questions 23-33
- Solution Sheet 3 Questions 34-49

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 all the Theorems nor solutions to all Examples. There has to be some reason for attendance at lectures.

Notes | Contents |
---|---|

Notes Part 1.1a
Notes Part 1.1b
Part 1.1 Appendix |
1.1 Limits limit of a real-valued function at a finite point, one-sided limits, limits of a function at infinity. Divergence. |

Notes Part 1.2a
Notes Part 1.2b
Part 1.2 Appendix |
1.2 Limits Limits Rules, including the Sum, Product and Quotient rules and the Sandwich Rule. |

Notes Part 1.3 Part 1.3 Appendix |
1.3 Special Limits. The exponential and trigonometric functions. |

Notes Part 2.1
Part 2.1 Appendix |
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
Part 2.2 Appendix |
2.2 Continuous functions Properties of continuous functions, including the Intermediate Value Theorem, and the Boundedness Theorem. |

Notes Part 2.3
Part 2.3 Appendix |
2.3 Continuous functions Monotonic functions, the Inverse Function Theorem. |

Notes Part 3.1a
Notes Part 3.1b
Part 3.1 Appendix |
3.1 Differentiation Definition. Rules for differentiation, including the Sum, Product and Quotient rules, the Chain Rule and the Inverse Rule. |

Part 3.2 Appendix | 3.2 Differentiation Derivative Results including: Rolle's Theorem, the Mean Value Theorem, Cauchy's Mean Value Theorem and L'Hôpital's Rule. |

Part 3.3 Appendix | 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. |

Part 4 Appendix |
4 Integration
Definition and examples. Fundamental Theorem of Integration.
Not given in 2016-17: How to use the Fundamental Theorem of Calculus; an example of a sequence of Riemann integrable functions whose limit is not Riemann integrable; if f is monotonic then f is Riemann integrable; Sum, Product and Quotient Rules for integration; Integration by parts; Integration by Substitution; Improper Integrals. |

I must stress that you cannot pass the exam simply by doing past papers. You can get practice of how long the questions are, learn how each part of a question depends on previous parts of that question, and get a feel of what types of problems you may be required to solve, definitions of give and Theorems to prove. But to pass exams you have to learn, understand and remember the material in the course. For this, nothing is better than attending the lectures along with reading the notes on this site.

- Mary Hart,
*Guide to Analysis*, Macmillan Mathematical Guides, Palgrave Macmillan; second edition 2001, - R Haggarty,
*Fundamentals of Mathematical Analysis*, Addison-Wesley, second edition, 1993.