M3M6AppliedComplexAnalysis

Lecture notes and course material for Applied Complex Analysis (2021) at Imperial College

See also previous lecture notes for previous courses: M3M6 Methods of Mathematical Physics and Applied Complex Analysis (2020)

Module information

Module guide

Discussion forum/Q&A

On Piazza, sign up here

Office hours

Fridays, 11:00-12:00, MS Teams

Problem classes

Thursdays in Weeks 2, 4, 6, 8 and 10; 11:00-12:00 or 2:00-3:00, MS Teams

Project

1. Project proposal due 26 Feb 2021
2. Project due 26 March 2021

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
3. Hao Hao, 2D Uniform Ideal Fluid Flow

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
3. E. Wegert, Visual Complex Functions: An Introduction with Phase Portraits, Birkhäuser, 2012
4. B. Fornberg & C. Piret, Complex Variables and Analytic Functions: An Illustrated Introduction, SIAM, 2019

Problem sheets, mastery material and revision questions

1. Problem Sheet 1 (Solutions)
2. Problem Sheet 2 (Solutions)
3. Problem Sheet 3 (Solutions)
4. Problem Sheet 4 (Solutions)
5. Problem Sheet 5 (Solutions)
6. Mastery material, Mastery Sheet, (Solutions)
7. Revision questions

Video lectures

Panopto

Lecture notes

0. Running Julia code
1. Visualising complex functions
2. Cauchy's integral formula and Taylor series
3. Laurent series and residue calculus
4. Analyticity at infinity
5. Applications of complex integration to real integrals
6. Trapezium rule, Fourier series and Laurent series
7. Matrix norms and matrix functions
8. Computing matrix functions via Cauchy's integral formula and the trapezium rule
9. Matrix exponentials and the (fractional) heat equation
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. Inverting logarithmic singular integrals and 2D electrostatic potentials
18. Orthogonal polynomials
19. Classical orthogonal polynomials
20. Orthogonal polynomials and differential equations
21. Orthogonal polynomials and singular integrals
22. Hermite polynomials
23. Riemann–Hilbert problems
24. Laurent and Toeplitz operators
25. Half-Fourier and Laplace transforms
26. The Wiener–Hopf method