Stefan LLEWELLYN SMITH
EBUII 574
x23475
http://mae.ucsd.edu/~sgls
This is the homepage for MAE207 Applications of complex analysis during
Spring Quarter 2008.
Last
updated: April 17, 2008.
Rationale
Many solution techniques in applied mathematics use complex variable
techniques, in many cases leading to closed-form but difficult-to-use
solutions. The development of computers in the last 50 years has led to
brute-force approaches to many of these problems. However modern
computer software for symbolic and technical computing like Maple and
Matlab can also be used to evaluate and generalize the analytical
solutions. In this class, I will cover some of these approaches using
actual examples and as much of the underlying theory as needed.
Prerequisites
Complex variable, linear algebra. Numerical analysis would be helpful
but may not be absolutely necessary.
Computer resources
Access to matlab is crucial. If required we will use the MAE computer
laboratories.
Grading (tentative)
20% Handing in one completed written-up exercise every Monday.
40% Literature review/coding project (in groups?)
50% Final project: to be developed in collaboration with me,
applying some
of these techniques.
Tentative syllabus (20 lectures; maybe this is too much material,
but I hope to produce lecture notes as we go)
Introduction and review of compl ex numbers
Finding zeros
Branch cuts
ODEs in the complex plane
Quadrature
Fourier transforms
FFT
Calculus of residues
Laplace transforms
Inversion of Laplace transforms
Method of steepest-descents
WKB
Cagniard-de Hoop method
Laplace's equation
Conformal maps
Elliptic functions
Chebyshev expansions
Orthogonal polynomials
Wiener-Hopf method
Matrix factorizations
Useful books (updated in notes)
Ablowitz & Fokas
Complex variables
Carrier, Krook & Pearson Functions of a complex
variable : theory and technique
Henrici
Applied and computational complex
analysis
Trefethen
Spectral methods in Matlab
Grading policy
I remind you of UCSD's policy
on academic integrity. I may rescale the components to arrive at the
final
grade.