
ANALYTICAL AND STOCHASTIC MATHEMATICAL METHODS FOR ENGINEERING
Prerequisites:

None 
Examination methods:

Final examination based on: written examination. 
Course unit contents:

1. Lebesgue Measure and Integral — Lebesgue measure and integral on R^d, limit theorems, dependence by parameters, reduction formula and change of variables. Introduction to abstract measure and integral.
2. Normed spaces — Concept of norm; uniform, L^1 and L^p norms; sequences, limits, completeness and Banach spaces. Concept of scalar product and Hilbert space; the space L^2; separable spaces and concept of orthonormal base.
3. Fundamentals of Complex Analysis — Power series in C and elementary analytic functions; concept of holomorphic function, CauchyRiemann equations, analyticity of holomorphic functions; isolated singularities; residue theorem and applications.
4. Introduction to Fourier Analysis — Fourier series: Euler formulas, convergence and sum (pointwise, uniform, L^2). Fourier transform: L^1 definition, convolution and approximate units, inversion formula. Schwarz functions and L^2 Fourier transform. Applications.
5. Introduction to probability — probability spaces, axioms of probability, conditional probabilities, independence of events.
6. Random variables (discrete and continuous) — definition, expectation and moments. Examples of random variables and applications, with a focus on Gaussian random variables.
7. Random vectors.
8. Characteristic function.
9. Convergence of random variables: weak, in probability, in L^p, almost sure.
10. The law of large numbers and the central limit theorem with applications.
11. Conditional expectation.
12. Martingales in discrete time. 

