Fourier series and Fourier transform. Introduction to functional analysis: linear operators on Hilbert spaces. Complex analysis: Cauchy theorem, Taylor and Laurent series, calculus of residues. Laplace transform. Distributions.
G. Cosenza, "Metodi Matematici della Fisica", Bollati Boringhieri.
L. Debnath e P. Mikusinski, "Hilbert Spaces with Applications", Elsevier.
G. Pradisi, "Lezioni di metodi matematici della fisica", Edizioni della Normale.
W. Rudin, "Real and Complex Analysis", McGraw-Hill.
Esercizi: M.L. Krasnov, A.I. Kiselev e G.I. Makarenko, "Funzioni di variabile complessa e loro applicazioni", MIR 1981.
R. Shakarchi, "Problems and solutions for complex analysis", Springer 1999.
M.R. Spiegel, "Analisi di Fourier con applicazioni alle equazioni alle derivate parziali", collana Schaum.
M.R. Spiegel, "Laplace transforms", collana Schaum.
Learning Objectives
Knowledge acquired:
Mathematical methods for solving problems of mathematical physics and mathematical formalism of Quantum Mechanics.
Skills acquired (at the end of the course):
Calculus of integrals with the residue method, Fourier and Laplace transforms, solution of differential equations.
Prerequisites
Courses required: Mathematical analysis I, Geometry.
Teaching Methods
Exposition on the blackboard
Type of Assessment
Written exam (test of ability in solving problems) duration 2 hours;
oral exam (test of the degree of comprehension of conceptual aspects of the course) duration 45 minutes.
Course program
Fourier series. Fourier transform in L1 and L2.
Hilbert spaces, linear operators on Hilbert spaces, spectra of operators.
Complex analysis: holomorphy, integration, Cauchy theorem, Taylor and Laurent series, calculus of residues, Jordan lemma. Laplace transforms.