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

Real Analysis elements:

Sequances and series of functions. Pointwise and uniform convergence. Integral of Lebesgue (hints).

Functional Analysis elements:

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

Fourier Series:

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

Holomorphic function elements:

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

Laplace transform:

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

Fourier Transform:

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

Applications to PDEs (if time is left):

Linear first order PDEs and tha method of characteristics. Classification of equations for linear partial derivatives of the second order. Generalities and examples of PDEs: Laplace equation, wave equation and heat equation. Applications of the Fourier transform to selected PDEs.

Readings/Bibliography

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

 

For the exercises:
See the pdfs published on the teacher's website; if the student carefully solves these exercises, he/she will not need further texts.

For the theory: it is enough that the students regularly follows ALL frontal lessons given in the classroom, and that he/she studies the theory on the notes taken during the lectures. The adopted textbook is an optional tool.

It is strongly recommended that non-attending students do obtain the notes taken by some regularly attending students. This will allow the non-attending student to save time and effort in preparing the written exam. Obviously, it is -however- a full right of the non-attending student to prepare the exam by also using the suggested texts.

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

ONLY written examination.

The written examination will check the knowledge of the topics presented in the exercises, regularly published on the dedicated Unibo site; in the written part, some questions may be posed on theoretical topics as well.

Dates:

3 exams in January/February

1 in June: 1 in July, 1 in September

PLEASE, use Almaesami to book for the examination dates!

Teaching tools

See the dedicated Unibo site.

Office hours

See the website of Andrea Bonfiglioli