CODE 106950 2023/2024 7 cfu anno 3 MATEMATICA 8760 (L-35) - GENOVA MAT/05 Italian GENOVA 1° Semester AULAWEB

## OVERVIEW

Course of the III year of Laurea Triennale in Mathematics; the course consists of two parts: complex analysis and functional analysis, both at introductory level.

## AIMS AND CONTENT

### LEARNING OUTCOMES

The objective of the course is to provide an introduction to three areas of Mathematical Analysis which are fundamental for their theoretical and applied implications and developments. Complex analysis: Cauchy-Riemann equations, Cauchy theorem and consequences, singularities and Laurent series, residue theorem. Hilbert spaces: normed spaces and bounded linear operators, Hilbert spaces and orthonormal bases, L1 and L2 spaces, Riesz representation and projection theorems. Fourier series: Fourier series in L ^ 2, L ^ 1, pointwise convergence.

### AIMS AND LEARNING OUTCOMES

Students will be able to solve simple problems and to follow more advanced studies in complex and functional analysis.

### PREREQUISITES

Analysis, Geometry and Algebra courses from the first two years of Laurea Triennale in Mathematics.

### TEACHING METHODS

The classical method: lectures and exercises using the blackboard; written (exercises) and oral (theory and exercises) examination. Special importance is given to exercises and consequently to the written part of the exam.

### SYLLABUS/CONTENT

Complex Analysis: power series and analytic functions; complex differentiation and holomorpgic functions; complex integration, Cauchy's theorem and primitives; classical consequences of Cauchy's theorem; singularities, residue theorem and applications.

Functional Analysis: normed spaces; linear operators; scalar products; Hilbert spaces and orthonormal bases; projection theorem and Riesz representation theorem; study of important examples: the space L^2.

Fourier Analysis: Fourier series in L^2. Fourier series in L^1: an outline of results on the convergence, punctual and in norm.

V.Villani - Funzioni di Una Variabile Complessa - Edizioni Scientifiche Genova 1971.

I.Stewart, D.Tall - Complex Analysis, 2nd ed. - Cambridge U. P. 2018.

H.Cartan - Elementary Theory of Analytic Functions of One or Several Variables - Dover Publ. 1995.

A.I.Markushevich - Theory of Functions of a Complex Variable, parts I--III - A.M.S. Chelsea Publishing 2005.

W.Rudin - Analisi Reale e Complessa - Bollati Boringhieri 1978.

M.Reed, B.Simon - Functional analysis - Academic Press 1972.

E.M.Stein, R.Shakarchi - Real Analysis - Princeton U. P. 2005.

## LESSONS

### LESSONS START

When lessons of the III year start.

### Class schedule

L'orario di tutti gli insegnamenti è consultabile all'indirizzo EasyAcademy.

## EXAMS

### EXAM DESCRIPTION

Written and oral examination.

Students with a certified DSA, disability or other special educational needs are advised to contact the lecturer at the beginning of the course in order to agree on teaching and examination methods that, while respecting the teaching objectives, take into account individual learning methods and provide suitable compensatory tools.

### ASSESSMENT METHODS

Evaluation of written and oral examination. In the written part, some exercises will be proposed, and the quality of the solutions written by the students will be evaluated. The oral part deals mainly with the theory developed during the course, and the understanding of the theorems and the ability of reproducing proofs of the students will be evaluated.

### Exam schedule

Data Ora Luogo Degree type Note
08/01/2024 09:00 GENOVA Scritto
10/01/2024 09:00 GENOVA Orale
29/01/2024 09:00 GENOVA Scritto
31/01/2024 09:00 GENOVA Orale
03/06/2024 09:00 GENOVA Scritto
05/06/2024 09:00 GENOVA Orale
02/07/2024 09:00 GENOVA Scritto
03/07/2024 09:00 GENOVA Orale
10/09/2024 09:00 GENOVA Scritto
11/09/2024 09:00 GENOVA Orale