M3M6AppliedComplexAnalysis

Lecture notes and course material for M3M6 Applied Complex Analysis at Imperial College

Office Hours: 11:00-12:00 Tuesdays 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

See also previous lecture notes for previous course M3M6 Methods of Mathematical Physics

Problem sheets and mastery material

1. Problem Sheet 1 (Solutions)
2. Problem Sheet 2 (Solutions)
3. Problem Sheet 3 (Solutions)
4. Problem Sheet 4 (Solutions)
5. Mastery material (Solutions)
6. Mastery Sheet

Project

1. Project proposal due 19 Feb 2020
2. Project due 19 March 2020

Examples of previous projects:

1. Wasim Rehman, Quantum Mechanics and Matrix Functions via Trapezium Rule
2. Tianyi Pu, 2D Ideal Fluid Flow Around an Obstacle

Lecture notes

1. Review of complex analysis
2. Cauchy's integral formula and Taylor series
3. Laurent series and residue calculus
4. Analyticity at infinity
5. Applications to real integrals
6. Convergence rate of the trapezium rule
7. Matrix norms and matrix functions
8. Matrix functions via Cauchy's integral formula
9. Computing matrix functions via the trapezium rule
10. Branch cuts
11. Representing analytic functions by their behaviour near singularities
12. Cauchy transforms and Plemelj's theorem
13. Hilbert transforms
14. Inverting the Hilbert transform and ideal fluid flow
15. Electrostatic charges in a potential well
16. Logarithmic singular integrals
17. Logarithmic singular integral examples
18. Inverting logarithmic singular integrals
19. Orthogonal polynomials
20. Classical orthogonal polynomials
21. Orthogonal polynomials and differential equations
22. Orthogonal polynomials and singular integrals
23. Hermite polynomials
24. Riemann–Hilbert problems
25. Laurent and Toeplitz operators
26. Half-Fourier and Laplace transforms
27. The Wiener–Hopf method