Introduction to functional analysis: operators on Hilbert spaces. Fourier transform. Distributions. Complex analysis: Cauchy theorem, Taylor and Laurent series, calculus of residues, Jordan lemma. Laplace transform.
M. Ablowitz and A. Fokas, "Complex Variables - Introduction and Apllications", Cambridge Texts in Applied Mathematics.
G. Cosenza, "Metodi Matematici della Fisica", Bollati Boringhieri.
L. Debnath and 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.
Exercises: 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", Schaum.
M.R. Spiegel, "Laplace transforms", Schaum.
Learning Objectives
Knowledge acquired:
Mathematical methods for solving problems of mathematical physics and mathematical formalism of Quantum Mechanics.
Competence acquired:
Complex Analysis and Functional Analysis.
Skills acquired (at the end of the course):
Calculus of integrals with the residue method, Fourier and Laplace transforms, use of distributions, solution of differential equations.
Prerequisites
Courses required: Mathematical analysis I, Geometry.
Teaching Methods
CFU: 6
Total hours of the course (including the time spent in attending lectures, seminars, private study, examinations, etc...): 150
Contact hours for: Lectures (hours): 52
Type of Assessment
Written and oral exam.
Course program
Hilbert spaces, linear operators on Hilbert spaces, spectra of operators, spectral theorem. Green functions, Sturm-Liouville problems.
Fourier transform in L1 and L2. Distributions, tempered distributions, Fourier transform of tempered distributions.
Complex analysis: holomorphy, integration, Cauchy theorem, Taylor and Laurent series, calculus of residues, Jordan lemma, analytic continuation.
Laplace transforms.