Course material for M3M6 Methods of Mathematical Physics

Office Hours: 16:00-17:00 Mondays during term, Huxley 6M40

Reading list

1. M.J. Ablowitz & A.S. Fokas, Complex Variables: Introduction and Applications, Second Edition, Cambridge University Press, 2003
2. R. Earl, Metric Spaces and Complex Analysis, 2015

Problem sheets, exams, and mastery material (2018)

1. Problem sheet 1 (Solutions)

2. Problem sheet 2 (Solutions)

3. Problem sheet 3 (Solutions)

4. Problem sheet 4 (Solutions)

5. Mastery sheet (Solutions)

6. Mastery material (Available on Blackboard)

7. Practice exam

Lecture notes (2018)

Each file is a Jupyter notebook, that can be viewed using the Jupyter Notebook viewer:

1. Course overview
2. Cauchy's theorem
3. Cauchy's integral formula and Taylor series
4. Trapezium rule, Fourier series and Laurent series
5. Residue theorem
6. Analyticity at infinity
7. Integrals over the real line
8. Functions with branch cuts
9. Cauchy transforms
10. Hilbert transforms
11. Riemann–Hilbert problems
12. Ideal fluid flow
13. Electric charges in a potential well
14. Constructing orthogonal polynomials
15. Recurrence relationships
16. Solving differential equations with orthogonal polynomials
17. Differential equations satisfied by orthogonal polynomials
18. Orthogonal polynomials and singular integrals
19. Logarithmic singular integrals
20. Integral equations on the real line
21. Laplace transforms
22. Integral equations on the half-line and Riemann–Hilbert problems
23. Cauchy transforms on the real line
24. Riemann–Hilbert problems on the real line

Lecture notes (2017)

25. The Wiener–Hopf method
26. Analyticity of solutions of ordinary differential equations
27. Singular points of ordinary differential equations
28. Hypergeometric functions

To run the files on your own machine, use the following steps:

1. Download Julia v1.1
2. In Julia, install the necessary packages:

   Pkg.add("ApproxFun")
   Pkg.add("Plots")
   Pkg.add("GR")
   Pkg.add("Plotly")
   Pkg.add("PlotlyJS")
   Pkg.add("Interact")
   Pkg.add("IJulia")
   Pkg.add("DifferentialEquations")
   Pkg.add("ComplexPhasePortrait")
   Pkg.clone("https://github.com/JuliaApproximation/OscillatoryIntegrals.jl")

3. Boot-up Jupyter by running in Julia

   using IJulia
   @async notebook()

4. Download the files and drag and drop (or better yet, use git to clone the reposaitory and stay up-to-date).