# MAGIC022: Mathematical Methods

## Course details

A core MAGIC course

### Semester

Autumn 2022
Monday, October 3rd to Friday, December 9th

### Hours

Live lecture hours
10
Recorded lecture hours
10
80

Thursdays
09:05 - 09:55

### Course forum

Visit the MAGIC022 forum

## 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

• MS

### Dr Mike Simon

University
University of Manchester

## Bibliography

No bibliography has been specified for this course.

## Assessment

The assessment for this course will be released on Monday 9th January 2023 and is due in by Sunday 22nd January 2023 at 23:59.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hoursâ€™ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.