Mathematical Methods Matteo Franca
Basic instrument of the mathematical analysis in one and more real variables, and of one complex variable. Olmorphic functions theory in one variable. Basic instruments of linear algebra
KNOWLEDGE AND UNDERSTANDING:
The student should know the theory of complex functions of one variable (limits, continuity, differentiability, intagration along paths), of basic properties of holomorphic functions, of Laplace and Fourier transform, as well as their applications to the solutions of concrete problems. Students should be able to apply notions and theorems to solve real problems.CAPACITY TO APPLY KNOWLEDGE AND UNDERSTANDING:
The student must develop the ability of solving concrete problems by applying theorems, tools and methods.TRANSVERSAL SKILLS:
The student will be able to critically evaluate different practical solutions in order to identify the most appropriate objectives to be pursued.
Lebesgue spaces. Fubini and Tonelli theorems, Lebesgue theorem.
Fourier series: meaning and evaluation of the coefficients. Bessel inequality, simple, uniform and L^2 convergence of the series.
Fourier transform. Algebraic and differential properties. Duality formula. Schwartz spaces and Plancherel theorem. Laplace transform and signals in complex fields. Convergence abscissa. Connection between Fourier and Laplace transform.
Linear partial differential equation. First order equation: Fourier and characteristics methods.
Second order PDE Elliptic, parabolic and hyperbolic
Development of the examination
LEARNING EVALUATION METHODS
The exam consists of a written and an oral part.
LEARNING EVALUATION CRITERIA
The exam is meant to verify the capability of solving exercise and the knowledge of theorical instrument for them useful, and to evaluate the skill in connecting the objects described in the course, and to use an appropriate technical language.
LEARNING MEASUREMENT CRITERIA
To access the oral part it is necessary to pass the written one. To pass the written test is necessary to get an evaluation equal or superior to 15 and equal or superior to 18 for the oral test.
To pass the tests the student needs to have understood the basic concepts introduced in the course, and their use to solve standard exercises. To reach a better evaluation he/she needs to know the proofs given in the course and to solve less standard exercises.
FINAL MARK ALLOCATION CRITERIA
The final mark is the result of written tests, modified by the evaluation of the oral test. If the student fails this latter test he has to redo the exam, and if the evaluation is negative the final mark could decrease, while if it is positive it could increase at most of 7 points.
In the web page at the following address
the student can find the main part of the topics discussed.
For the part concerning Fourier and Laplace transform we suggest the book
G. C. Barozzi, "Matematica per l'Ingegneria dell'Informazione", Zanichelli, Bologna, 2001.
- Ingegneria Elettronica (Corso di Laurea Triennale (DM 270/04))