29145 - Integrational Elements of Mathematical Analysis L

Learning outcomes

The course will enable the students to learn the basic mathematical tools concerning with Fourier series and Fourier and Laplace transforms (via the knowledge of some basic functional and complex Analysis); applications to ODEs and PDEs will also be considered.

Course contents

Functional Analysis:

Metric spaces. Banach spaces. Hilbert Spaces. Space L^2 (hints). Orthonormal bases.

Fourier Series:

Main definitions; Fourier series convergence theorems for a periodic function.

Laplace transform:

Main definitions concerning the Laplace transform; Convergence domain. Calculation rules. Antitransform. Application to Linear ODEs.

Holomorphic function:

The Cauchy Theorem; Laurent series; the Residue Theorem and the calculation of some integrals using the Residue Theorem.

Fourier Transform:

Main definitions relating to the Fourier transform of an L^1 function (and references to the L^2 case). Calculation rules. Antitransform.

Readings/Bibliography

G.C. Barozzi, Matematica per l'informazione, Zanichelli

Teaching methods

The course includes theoretical lessons (in which the first elements of complex and functional analysis, the Laplace and Fourier transforms and the Fourier series) will be introduced, together with exercises aimed to help the students gain familiarity with the tools and mathematical methods introduced during the lessons.

Assessment methods

The exam consists ONLY of a written examination in whichthe student will be asked to solve a few exercises and to answer to theoretical quetsions.

Note: the student is allowed to to register to a written examination even in case of failure in the previous exam.

Dates: 3 exams in January/February, 1 in June, 1 in July and 1 in September.

Use Almaesami to book for the examination dates and pay attention to register in the proper time.

Office hours

See the website of Eugenio Vecchi