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 goal is to provide students with the fundamental principles and techniques of complex analysis and functional analysis, with particular focus on applications to Fourier series. These tools represent an essential step toward mastering techniques and concepts that appear across many areas of both theoretical and applied mathematics, and are crucial for engaging meaningfully with more advanced studies and research.

AIMS AND LEARNING OUTCOMES

The aim of the course is to introduce the basic tools of complex analysis and functional analysis, along with their related techniques and applications. Fourier series will also be briefly introduced by applying concepts from functional analysis. By the end of the course, students are expected to be able to solve simple problems in Complex and Functional Analysis at a level sufficient to pursue more advanced studies in Analysis.

PREREQUISITES

Courses in Analysis from the first two years of the Laurea Triennale and basic concepts of algebra and topology.

TEACHING METHODS

The classical method: lectures using the blackboard. Special importance is given to examples and exercises.

SYLLABUS/CONTENT

Complex Analysis: power series and analytic functions; complex differentiation and holomorphic 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 and some results on pointwise convergence.

RECOMMENDED READING/BIBLIOGRAPHY

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.

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

September 22, 2025

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Written and Oral exam. Students with a grade greater than or equal to 14 are admitted to the oral exam. The final grade is based on the grade of the written exam, but can be increased or decreased depending on the student's performance in the oral exam.

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.

FURTHER INFORMATION

Students with disabilities or specific learning disorders (DSA) are reminded that, in order to request exam accommodations, they must first upload the relevant certification to the University website at servizionline.unige.it, in the “Students” section. The documentation will be verified by the University's Office for Inclusion Services for Students with Disabilities and SLD.

Subsequently, well in advance (at least 10 days) before the exam date, students must send an email to the instructor responsible for the exam, copying both the School's contact person for the inclusion of students with disabilities and DSA (sergio.didomizio@unige.it), and the above-mentioned University office. The email must include the following information:

Course title

Exam date

Student’s last name, first name, and student ID number

The requested compensatory tools and dispensatory measures deemed necessary

The inclusion contact person will confirm to the instructor that the student is entitled to request accommodations for the exam, and that such accommodations must be agreed upon with the instructor. The instructor will then inform the student whether the requested measures can be applied.

Requests must be sent at least 10 days before the exam date to allow the instructor sufficient time for evaluation. In particular, if concept maps are intended to be used during the exam (which must be much more concise than those used for study purposes), failing to respect the deadlines may make it impossible to guarantee enough time for any necessary adjustments.

For further information on requesting services and accommodations, please refer to the document “Guidelines for requesting services, compensatory tools and/or dispensatory measures, and specific aids.”