Anno di corso: 1

Anno di corso: 2

Crediti: 12
Tipo: A scelta dello studente
Crediti: 30
Tipo: Lingua/Prova Finale
Crediti: 3
Tipo: Altro
Crediti: 3
Tipo: Altro

FUNCTIONAL ANALYSIS

Scheda dell'insegnamento

Anno accademico di regolamento: 
2018/2019
Anno di corso: 
1
Anno accademico di erogazione: 
2018/2019
Tipo di attività: 
Obbligatorio
Lingua: 
Inglese
Crediti: 
6
Ciclo: 
Primo Semestre
Ore di attivita' didattica: 
52
Prerequisiti: 

Basic mathematical analysis: differential calculus for functions of one or several variables, ordinary and partial differential equations, integral calculus.

Moduli

Metodi di valutazione

Tipo di esame: 
Orale
Modalita' di verifica dell'apprendimento: 

Written exam: exercises and problems with open questions.

Oral exam: discussion of the written exam; possible request for resolution of other exercises; questions on definitions, statements and proofs of theorems.

It is possible to take the oral exam even if the result of the written exam is not sufficient.

Valutazione: 
Voto Finale

Obiettivi formativi

The aim of the course is to provide the basic tools of Mathematical Analysis necessary for the study of the differential equations of quantum and classical Mechanics and of Physics in general.

Contenuti

Complex analysis. Special functions. Fourier series. Convolution. Fourier transform. Distributions. Laplace transform. Elements of Calculus of Variations.

Programma esteso

Complex Analysis

Holomorphic functions and harmonic functions. Cauchy's theorem. Laurent series. Residue theorem. Lemma of Jordan. Calculation of integrals applying the residual theorem.

Fourier series

Serial development compared to a complete orthonormal system. Parseval formula and inversion formula. Fourier series in real and complex form.

Fourier transform and applications

Parseval formula and inversion formula. Convolution of functions. Applications to the resolution of the heat equation and the wave equation. Calculation of Fourier transforms with the residual theorem. Gaussian function. Lorentzian function. Voigt function. Distribution of Fourier transforms. Approximation of the Dirac delta.

Special functions

Functions of Laguerre, Legendre, Bessel. Spherical harmonics.

Elements of Calculus of Variations

Functional derivative. Euler-Lagrange equation.

Laplace transform

Application of the Laplace transform to the resolution of ordinary differential equations with edge conditions.

Bibliografia consigliata

K. F. Riley, M. P. Hobson and S. J. Bence. Mathematical Methods for Physics and Engineering, Cambridge University Press.

Modalità di erogazione

Convenzionale

Metodi didattici

Lectures and exercises.