# MAGIC022: Mathematical Methods

## Course details

A core MAGIC course

### Semester

Autumn 2013
Monday, October 7th to Saturday, December 14th

### Hours

Live lecture hours
20
Recorded lecture hours
0
0

Mondays
12:05 - 12:55
Thursdays
10:05 - 10:55

## Description

This is a core applied module. The aim of the course is to pool together a number of advanced mathematical methods which students doing research (in applied mathematics) should know about. Students will be expected to do extensive reading from selected texts, as well as try out example problems to reinforce the material covered in lectures. A number of topics are suggested below and depending on time available, most will be covered. The course proceeds at a fairly fast pace.
Assessment The assessment for this module will be in the form of a take-home exam at the end of the course.

Recommended books:
• Bender and Orsag, Advanced mathematical methods for scientists and engineers
• Bleistan and Handlesman, Asymptotic expansions of integrals
• Hinch, Perturbation methods
• Ablowitz & Fokas Complex Variables, C.U.P.
• Lighthill Generalised Functions, Dover paperback.

### Prerequisites

It is assumed that students have done some real and complex analysis.

### Syllabus

• Advanced differential equations, series solution,classification of singularities. Properties near ordinary and regular singular points. Approximate behaviour near irregular singular points. Method of dominant balance. Airy, Gamma and Bessel functions.
• Asymptotic methods. Boundary layer theory. Regular and singular perturbation problems. Uniform approximations. Interior layes. LG approximation, WKBJ method.
• Generalised functions. Basic definitions and properties.
• Revision of basic complex analysis. Laurent expansions. Singularities. Cauchy's Theorem. Residue calculus. Plemelj formuale.
• Transform methods. Fourier transform. FT of generalised functions. Laplace Transform. Properties of Gamma function. Mellin Transform. Analytic continuation of Mellin transforms.
• Asymptotic expansion of integrals. Laplace's method. Watson's Lemma. Method of stationary phase. Method of steepest descent. Estimation using Mellin transform technique.
• Conformal mapping. Riemann-Hilbert problems.

## Lecturer

• JG

### Prof Jitesh Gajjar

University
University of Manchester

## Bibliography

### Follow the link for a book to take you to the relevant Google Book Search page

You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library' - this sometimes works well, but not always - you will need to enter your location, but it will be saved after you do that for the first time.

## Assessment

### Description

Assessment for this course will be via a take-home examination which will be put online after the end of the course at the beginning of January. The exam paper will require the completion of 6/6 questions and to pass one is required to obtain at least 50%. The deadline for the work to be completed is midnight, 19th January 2014.

### Assessment not available

Assessments are only visible to those being assessed for the course.