Course Syllabus
Obiettivi
Contenuti sintetici
Programma esteso
Prerequisiti
Modalità didattica
Materiale didattico
Periodo di erogazione dell'insegnamento
Modalità di verifica del profitto e valutazione
Orario di ricevimento
Aims
The aim of the course is to provide the basic tools of Mathematical Analysis useful in the study of the differential equations of Classical Physics and Quantum Mechanics.
Contents
Complex analysis. Special functions. Fourier series. Convolution. Fourier transform. Distributions and Dirac delta. Laplace transform. Elements of Calculus of Variations.
Detailed program
Complex Analysis
Holomorphic functions and harmonic functions. Cauchy's theorem. Laurent series. Residue theorem. Jordan Lemma. Calculation of integrals by means of residue theorem.
Fourier series
Reminder on series expansion. Complete orthonormal systems. 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, wave and Schroedinger equation. Calculation of Fourier transforms with the residue theorem. Gaussian function. Lorentzian function. Voigt function. Distribution of Fourier transforms. Distributions. Definition and simple properties. Approximation of the Dirac delta distribution.
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 solution of ordinary differential equations with boundary or initial conditions.
Prerequisites
Basic mathematical analysis: differential calculus for functions of one or several variables, ordinary and partial differential equations, integral calculus.
Teaching form
Lectures and exercises.
Textbook and teaching resource
K. F. Riley, M. P. Hobson and S. J. Bence. Mathematical Methods for Physics and Engineering, Cambridge University Press.
Further material will be suggested or distributed during the course.
Semester
First Semester 2018-19
Assessment method
Written exam: exercises and problems with open questions.
Oral exam: discussion of the written exam; solution of further exercises can be required; questions on definitions, statements and (selected) proofs of theorems.
It is possible to take the oral exam even if the result of the written exam is not sufficient.
Office hours
By appointment.
Students' opinion
Bibliography
Staff
- Diego Davide Noja