Course contents: Mathematical Methods 2008
(This is a preliminary list and may be modified as the course
progresses.)
- Standard differential equations: usual suspect solutions,
introduction to Mathematica for visualization
- Vector and tensor calculus: vector space, metric, differential
operators in general coordinates, Gauss's and Stokes' theorems
- Linear algebra:
matrices as operators, diagonalization, eigenvalues,
eigenvectors, eigenfunctions
- Series of numbers and functions: absolute, uniform and
asymptotic convergence, power series
- Complex analysis: analyticity, Cauchy's integral theorem,
Laurent expansion, singularities, analytic continuation,
calculus of residues, evaluating integrals
- Approaches for solving linear differential equations:
separation of variables, series solutions, Green's function,
Fourier and Laplace transforms
- Sturm-Liouville theory: functions as infinite dimensional
vector spaces, orthogonal basis, eigenvalue problems
- Special functions: generic properties in the light of
Sturm-Liouville theory and complex analysis
- Essential statistics: Bayes' theorem, binomial - poisson -
gaussian distributions, central limit theorem, data fitting,
hypothesis testing