Mathematical Methods in Physics

Aim of the course is to introduce the students to the methods of complex analysis and functional analysis. In spite of its introductory level, the course also tries to be rigorous, and many proofs are included when significant. Very important points of the program are:
-Holomorfic functions with examples of maps, Taylor series, the Cauchy theorem and its use, isolated singurities and Laurent's expansion, the residue theorem and integration in the complex plane. Analytic continuation.
- Banach and Hilbert spaces, examples of spaces of functions. Introduction to the theory of linear operators on Hilbert spaces.
-Fourier series and Fourier and Laplace transforms.
-Introduction to the theory of tempered distributions.

At the end of the course the student will be able to:
1) use and manipulate complex numbers along with their geometric meaning and representation, carry out arithmetic and algebraic operations in the complex plane, study geometric mappings.
2) will be able to carry out studies of functions (single and multi-valued) in the complex plane
3) will be able to compute integrals in the complex plane, with integration techniques based upon Cauchy's theorem and the calculus of residues
4) will be able to understand and utilize basic concepts about Hilbert and Banach spaces, and orthonormal functions (Hermite, Legendre)
5) will be able to understand the main properties of linear bounded operators such as projections, isometries, unitary operators, functions of an operator, self-adjointness (with the extension to unbounded operators). Will be able to apply the theory to finite matrix operators, and in some infinite-dimensional cases.
6) will have knowledge of Fourier series, their point and norm convergence, and evaluate the series for simple functions.
7) will have knowledge of the Fourier (and Laplace) transform in L1 and L2, and of Riemann-Lebesgue's theorem. He will evaluate the main Fourier transforms, also by techniques of integration in the complex plane.
7) will have knowledge of the basic theory of tempered distributions, the most important ones (delta, theta, principal part), their
derivative and Fourier transform, and applications (Sokhotskii-Plemelj identity).