MATHEMATICAL METHODS FOR PHYSICS

Knowledge of the fundamentals of complex analysis and functional analysis, with applications to physics (specifically: quantum mechanics and special relativity).

Lectures (taught classes), and guided exercises (in class).

Fundamentals of complex analysis:

- Complex plane, complex functions of complex variable, analytic functions, Cauchy-Riemann conditions, conformal transformations;
- Line integral, Cauchy theorem, Morera theorem, Cauchy formula;
- Series of functions, Weierstrass theorem, Cauchy-Hadamard theorem;
- Taylor series, Laurent series, singularities, residue theorem, integrals and series by the residue theorem;
- Fourier series, integral transforms, theory of distributions;
- Advanced topics: analytic continuation, asymptotic expansion.

Fundamentals of functional analysis:

- Reminder of linear algebra, metric spaces, linear spaces, normed spaces and Banach spaces, Euclidean spaces, separable Euclidean spaces, complete Euclidean space, Hilbert spaces, subspaces and orthogonal complement, linear functionals, Riesz theorem;
- Linear operators, continuous operators, adjoint of an operator, spectrum of an operator, compact and self-adjoint operators, functions of an operator, unitary operators;
- Finite-dimensional spaces, basis change, eigenvalue problem, diagonalization, series and functions of matrices;
- Group theory (an introduction): irreducible representarions, Lie groups, generators and algebra thereof, representations of SO(3), SU(2), SU(3), L(4), with examples from physics.

Recommended textbooks:
C. Presilla, Elementi di analisi complessa (Springer, Milano, 2014).
C. Bernardini, O. Ragnisco, P.M. Santini, Metodi matematici della Fisica, Carocci Ed.
M. R. Spiegel, Variabili Complesse, Etas Libri
G. G. N. Angilella, Esercizi di Metodi Matematici della Fisica (Springer, Milano, 2011)
G. Di Fazio, M. Frasca, Metodi matematici per l'ingegneria (Monduzzi, Bologna, 2009).
P. A. Grassi, Esercizi di metodi matematici per fisici e ingegneri (CEA, 2018)

A.N. Kolmogorov, S.V. Fomin, Elem. di teoria delle funz. e di anal. funzionale (Mir)
G. Fonte, Appunti di metodi matematici della fisica (Carocci, 2018)
G. Fano, Metodi matematici della meccanica quantistica (Zanichelli).
G. Cosenza, Metodi Matematici della Fisica, Bollati Boringhieri.
F. Bagarello, Fisica Matematica, Zanichelli 2007.
G. Cicogna, Metodi matematici della Fisica, Springer-Verlag Italia 2008.