# MAGIC022: Mathematical Methods

## Course details

A core MAGIC course

### Semester

Autumn 2020
Monday, October 5th to Friday, December 11th

### Hours

Live lecture hours
7
Recorded lecture hours
13
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.

## Lecturers

• MS

### Dr Mike Simon

University
University of Manchester
Role
Main contact
• AT

### Dr Alice Thompson

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 will be via a Take-Home exam. You will be given 4 or 5 questions to do (100 marks in total) and to pass the course you need 50%.

### Assessment not available

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

## Files

Files marked Lecture are intended to be displayed on the main screen during lectures.

File Associated lecture(s)
video 1a.mp4 Lecture
video 1b.mp4 Lecture
video 1c.mp4 Lecture
video 2a.mp4 Lecture
video 2b.mp4 Lecture
video 2c.mp4 Lecture
video 3a.mp4 Lecture
video 3b.mp4 Lecture
video 3c.mp4 Lecture
video 4a.mp4 Lecture
video 4b.mp4 Lecture
video 4c.mp4 Lecture
MATH64051-MAGIC022 Examples 1.pdf
video 5a.mp4
video 5b.mp4
video 5c.mp4
video 6a.mp4
video 6b.mp4
video 6c.mp4
MATH64051-MAGIC022 Examples 2.pdf
video 7a.mp4
video 7b.mp4
video 7c.mp4
video 8a.mp4
video 8b.mp4
video 8c.mp4
video 8d.mp4
MATH64051-MAGIC022 Examples 3.pdf
MATH64051-MAGIC022 Solutions 1.pdf
MATH64051-MAGIC022 Solutions 2 (questions 1 & 2).pdf
Lecture slides for week 1.pdf
Lecture slides for week 2.pdf
Lecture slides for week 3.pdf
Lecture slides for week 4.pdf
MAGIC022 Complex Analysis Questionnaire.docx
video 9a.mp4
video 9b.mp4
video 9c.mp4
videob 10a.mp4
videob 10b.mp4
videob 10c.mp4
videob 10d.mp4
MATH64051-MAGIC022 Solutions 3 (questions 1 and 2).pdf
videob 11a.mp4
videob 11b.mp4
videob 11c.mp4
videob 12a.mp4
videob 12b.mp4
videob 12c.mp4
On ODEs solved near regular singular points.pdf
Handout on Fourier Transforms.pdf
Chapter 7start.pdf
Figure 7.1.pdf
Figure 7.2.pdf
Figure 7.3.pdf
video 13a.mp4
video 13b.mp4
video 13c.mp4
video 13d.mp4
video 14a.mp4
video 14b.mp4
video 14c.mp4
video 14d.mp4
Myslides13.pdf
Myslides14.pdf
MAGIC022 Examples 5.pdf
MATH64051-MAGIC022 Solutions 3.pdf
MATH64051-MAGIC022 Examples 4.pdf
video 15a.mp4
video 15b.mp4
video 15c.mp4
video 16a.mp4
video 16b.mp4
video 16c.mp4
Myslides15.pdf
Myslides16.pdf
MATH64051-MAGIC022 Solutions 4.pdf
MATH64051-MAGIC022 Solutions 2.pdf
MAGIC022 Solutions 5.pdf