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 aim is to give some basic tools in complex and functional analysis. Such tools will allow students to absorb some concepts and techniques fundamental for any direction of further study or job.

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

Please see here.

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

Compensatory and dispensatory measures Disability/Invalidity/Specific Learning Disorder

Dispensatory measures and compensatory tools are intended to enable students to achieve the same learning objectives as their fellow students, not to facilitate the examination.

The use of compensatory tools and the application of dispensatory measures must be authorised in advance by the teacher in agreement with the Referee.

To take advantage of the adaptations during the examination, fill in the Adaptation request form; the request will be automatically sent by the system to the teacher in charge of the teaching, to the Contact Person of your School/Area/Department and in copy to the Sector; you will also receive a copy of the request sent by e-mail.

The adjustments available to students are as follows:

• Additional time (+30% DSA)
• Additional time (+50% disability/invalidity)
• Additional time during oral exams to organise the answer
• Calculator (programmable and graphing calculators are not allowed)
• Conceptual Maps
• Tables and/or Forms
• Take the exam in written form
• Take the exam in oral form
• Tutor reader (for written tests only)
• Tutor-writer (for written tests only)

Your request for adaptations must be submitted at least 7 working days before the scheduled exam date.

All information for students with disabilities and DSA is available on the webpage: Services for students with disabilities or DSA | UniGe | University of Genoa

Reference for inclusion: Sergio Di Domizio - sergio.didomizio@unige.it