# MATHEMATICAL METHODS FOR PHYSICS

## Learning outcomes of the course unit

First goal is to complete a basic preparation in classical real and complex analysis, with the theory of analytic functions. Main part of the course is devoted to the theory of linear operators in finite dimensional spaces, with the due insight in algebra and metric topology. Tthe extension to functional spaces L1 and L2 will be obtained through approximation problems, orthogonal functions, Fourier series and transform, presented in the perspective of a first approach to quantum mechanics.

In this frame, there will be a short course on differential equations in the complex field, with applications to the Schroedinger equation.

## Prerequisites

Basic notions of real analysis, calculus, geometry and algebra

## Course contents summary

Numerical Fields.

Complex Analysis. Analytic functions, basic instruments.

Residues, power series, definite integrals.

Linear manifolds, abstract vector spaces. Linear dependence. Dimension.

Real and compolex spaces. Isomorphism.

Scalar product. Orthogonality.

Metric spaces. Basic notions in topology.

Basis, orthogonal systems, orthogonalization.

Basis transformation.

Linear functional and Riesz Theorem Dirac formalism.

Sequences and convergence.

Linear applications and matrices.

Abstract linear operators.

Diagonalization.

Adjoint operators.

Eigenvalues and eigenvectors.

Hermitian, unitary and normal operators.

Projectors. Function of operators.

Complete sets of hermitian operators.

Polynomials and orthogonal functions.

Approximation. L1 and L2 spaces.

Fourier series and transforms.

Differential equations in the complex field.

Special functions.

Applications to the Schroedinger equation

## Course contents

Numerical Fields.

Complex Analysis. Analytic functions, basic instruments.

Residues, power series, definite integrals.

Linear manifolds, abstract vector spaces. Linear dependence. Dimension.

Real and compolex spaces. Isomorphism.

Scalar product. Orthogonality.

Metric spaces. Basic notions in topology.

Basis, orthogonal systems, orthogonalization.

Basis transformation.

Linear functional and Riesz Theorem Dirac formalism.

Sequences and convergence.

Linear applications and matrices.

Abstract linear operators.

Diagonalization.

Adjoint operators.

Eigenvalues and eigenvectors.

Hermitian, unitary and normal operators.

Projectors. Function of operators.

Complete sets of hermitian operators.

Polynomials and orthogonal functions.

Approximation. L1 and L2 spaces.

Fourier series and transforms.

Differential equations in the complex field.

Special functions.

Applications to the Schroedinger equation.

## Recommended readings

V. Smirnov, Corso di Matematica superiore, vol.III,2 (MIR)

E. Onofri, Teoria degli Operatori lineari, http://www.fis.unipr.it/home/enrico.onofri/#Lezioni

F.G.Tricomi, Metodi Matematici della Fisica (Cedam)

M.Spiegel, Variabili Complesse (Schaum, Etas)

E.Kolmogorov, S.Fomin, Elementi di teoria delle funzioni e dell'analisi funzionale (ER)

## Teaching methods

Lessons and exercises in the classroom.

## Assessment methods and criteria

Final written and oral tests.

The written test consists in exercises aiming to check the skill in calculus, trating problems which are variations of exercises already developed in the lessons.

The oral test consists in the discussion of some typical subjects, showing the methodological and conceptual mastership on the fundamental topics.