MAGIC022: Mathematical Methods

Course details

A core MAGIC course


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


Live lecture hours
Recorded lecture hours
Total advised study hours


09:05 - 09:55

Course forum

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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. 


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


  • 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.


  • MS

    Dr Mike Simon

    University of Manchester
    Main contact
  • AT

    Dr Alice Thompson

    University of Manchester


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

File Associated lecture(s)
video 1a.mp4 Lecture
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video 1c.mp4 Lecture
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MATH64051-MAGIC022 Examples 1.pdf
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MATH64051-MAGIC022 Examples 2.pdf
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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
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MATH64051-MAGIC022 Solutions 3 (questions 1 and 2).pdf
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On ODEs solved near regular singular points.pdf
Handout about Integral Transforms.pdf
Handout on Fourier Transforms.pdf
Chapter 7start.pdf
Handout about Fourier's Inversion Formula.pdf
Figure 7.1.pdf
Figure 7.2.pdf
Figure 7.3.pdf
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MAGIC022 Examples 5.pdf
MATH64051-MAGIC022 Solutions 3.pdf
MATH64051-MAGIC022 Examples 4.pdf
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MATH64051-MAGIC022 Solutions 4.pdf
MATH64051-MAGIC022 Solutions 2.pdf
MAGIC022 Solutions 5.pdf


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