# MATHEMATICAL PHYSICS

## Learning outcomes of the course unit

We expect that at the end of the course the students will have to following abilities:

- to know useful tools to face problems of Mathematical Physics;

- to be able to present in a clear way, with a correct language from the mathematical and physical points of view, the main arguments of the course;

- to be able to solve different kinds of partial differential equations appearing in problems of Mathematical Physics, owing to methods studied in the course for the classical equations.

## Prerequisites

Knowledge of topics of mathematical courses of class L-35 (first degree).

## Course contents summary

Functions of a complex variable.

Fourier and Laplace transforms.

Green function and Sturm-Liouville problems.

Classification of second order PDEs.

Fundamental equations of mathematical physics: Laplace equation, heat equation, wave equation.

Conservation laws.

## Course contents

Functions of a complex variable: Cauchy-Riemann conditions, types of singularities, residues and integration, Laurent series.

Fourier and Laplace transforms: definitions and basic properties, transforms of the fundamental functions, transform of Dirac delta, inverse transforms.

Introduction to the Green function and to Sturm-Liouville problems.

Classification of linear second order PDEs with two independent variables; Cauchy problem.

Fundamental equations of mathematical physics: Laplace equation, heat equation, wave equation (physical derivation, mathematical properties, methods of solution).

Conservation laws.

## Recommended readings

F. Gazzola, F. Tomarelli, M. Zanotti, Analisi complessa - Trasformate - Equazioni Differenziali, Esculapio, Milano.

S. Salsa, Equazioni a derivate parziali, Springer, Milano.

G. Spiga, Problemi matematici della Fisica e dell'Ingegneria, Pitagora, Bologna.

A. N. Tichonov, A. A. Samarskij, Equazioni della fisica matematica, MIR, Mosca.

## Teaching methods

During class lectures, the topics will be proposed from a formal point of view, and equipped with meaningful examples and applications.

Video of all lectures and pdf files of the slides written during the lectures will be uploaded on the web-site Elly.

## Assessment methods and criteria

The examination is based on an oral discussion, where the level of knowledge and understanding of the topics is valued, as well as the mathematical accuracy of exposition.