F1.3YQ1 Asymptotic analysis

(Third year module given in the Autumn term.)


Lecturer. Simon Malham, Room CM T.21, Mathematics Department.

Contact. email: simonm@ma.hw.ac.uk and tel: 0131 451 3254.

Lectures. Monday, Tuesday and Friday at 1:15pm.

Tutorials. One a week: Tuesday at 2:15pm.

Webpages. This course homepage where you can download lecture notes, further handouts, solutions to the exercises, Maple or Mathematica worksheets and other stuff is:


More information about the module can be found at:


which can also be reached from the Mathematics Department's Homepage --> Teaching --> Information for students already on course --> Modules mainly taken by Mathematics and AM&S students.

Course aims and objectives. To introduce techniques of asymptotic approximation for the solution of algebraic and differential equations and for the evaluation of integrals.


Assessment. The continuous assessment consists of a one hour midterm exam counting for 15 percent of the final mark (date to be announced) and a two-hour final exam at the end of term which counts for 85 percent of the course mark.

There is a resit in August/September for the ordinary course. The resit assessment is purely on the basis of a two-hour exam.

Calculators. In the final exam you will only be allowed to use either the Casio fx-85WA or fx-85MS. This is a University regulation. Personally, I do not think you will need a calculator in the Asymptotic Analysis exam.

Contract. Students are expected to read the notes in this booklet before, during and after the lectures and tutorials. Lectures will act as a more formal forum for the lecturer to explain the ideas of the course and give alternative examples, whilst tutorials will take a less formal and more personal form. There are starred exercises on problem sheets 1--7 and students must attempt these. Mathematics is best learned through grappling with the underlying ideas presented in lectures and then tackling problems given in the exercises.

You cannot learn to swim by reading a book about it!

Hence try the exercises, and if you get stuck, ask the lecturer either after a lecture, during the tutorials or during the office hours. It is vital that you can solve problems proficiently. If you need help, then

Ask, ask, ask!

Office hours. Once a week; we'll try to agree on these in the first week of term.

Attendance sheets. Students will be required to sign an attendance sheet with their initials in every lecture and tutorial. If any one student misses three consecutive such contact events, or more than one-third of them overall up until that date, then their personal mentor will be contacted.

Evaluations. At the end the course students will have an opportunity to fill out formal university evaluations on the course.

Books. The two main recommended books are Bender and Orszag and Hinch (see the bibliography of the main notes for details).


Electronic resources

PDF files are recommended for printing. denotes files not yet available.

Syllabus for 2004 (from the official department module pages)

Lecture notes.

Series and power series
Asymptotic analysis
Maple worksheet (.mws Maple file)
Heat equation
General polynomial and transcendental equations (non-examinable)
Chemical reaction kinetics (non-examinable)

Problem sheets and solutions. The solutions will be made available as the course progresses.

Topic Date due Solutions
Taylor series and order notation 14th Oct Solutions 1
Regular perturbation problems 21st Oct Solutions 2
Singular perturbation problems 28th Oct Solutions 3
Asymptotic series 4th Nov Solutions 4 2nd part
Laplace integrals 11th Nov Solutions 5
Method of stationary phase 18th Nov Solutions 6
Modelling 25th Nov Solutions 7

Movies. These are the matlab source files for the movies shown during the course. Download them and use them freely.

Taylor series approximations of sin(x)
Taylor series approximations of log(x)
Asymptotic approximations of the roots of a quadratic function
Asymptotic approximations of the roots of a cubic function
Laplace's method
Method of stationary phase

Exam papers. Hardcopies of solutions for the December 2001 -- December 2003 papers can be obtained from me in SRG04. We will discuss these past papers in week 9 during the usual lecture times.

Module timetable

This may change slightly as we progress so keep checking this webpage.

Lecture Section Topic
Oct 1st - Series revision
Oct 2nd - Power series revision
Oct 5th 1 Order notation
Oct 8th 2.1 Regular perturbations (expansion method)
Oct 9th 2.1 Regular perturbations (iteration method)
Oct 12th 2.1 Regular perturbations (non-integral powers)
Oct 15th 2.2 Singular perturbations
Oct 16th 2.2 Singular perturbations (transcendental equation)
Oct 19th 3.1 Asymptotic series
Oct 22nd 3.2/3.3 Asymptotic expansions and Properties of AEs
Oct 23rd 3.3 Properties of AEs (continued)
Oct 26th 3.4 Asymptotic expansions of integrals
Oct 29th 4/4.1 Laplace's method
Oct 30th 4.1 Laplace's method and Stirling's formula
Nov 2nd 4.2 Watson's lemma
Nov 5th 5 Method of stationary phase
Nov 6th 5 Method of stationary phase (continued)
Nov 9th - Application
Nov 12th - Mathematical modelling 1
Nov 13th - Mathematical modelling 2
Nov 16th - Mathematical modelling 3
Nov 19th - Mathematical modelling 4
Nov 20th - Mathematical modelling 5
Nov 23rd - Mathematical modelling 6
Nov 26th - Revision and past papers
Nov 27th - Revision and past papers
Nov 30th - Revision and past papers

This webpage and it's content was started on 5/10/2004.

Please feel free to download and use any of the material accessible from this page---provided that it is not used for commercial gain.

Last updated: 23/9/2008.