PORTALE DELLA DIDATTICA

### Mathematical methods

04LSILM, 02LSILP, 04LSIOD

A.A. 2020/21

Course Language

Inglese

Course degree

1st degree and Bachelor-level of the Bologna process in Ingegneria Informatica (Computer Engineering) - Torino
1st degree and Bachelor-level of the Bologna process in Electronic And Communications Engineering (Ingegneria Elettronica E Delle Comunicazioni) - Torino
1st degree and Bachelor-level of the Bologna process in Ingegneria Fisica - Torino

Course structure
Teaching Hours
Lezioni 70
Esercitazioni in aula 30
Teachers
Teacher Status SSD h.Les h.Ex h.Lab h.Tut Years teaching
Como Giacomo Professore Ordinario ING-INF/04 40 5 0 0 4
Teaching assistant
Context
SSD CFU Activities Area context
MAT/05
MAT/06
6
4
A - Di base
A - Di base
Matematica, informatica e statistica
Matematica, informatica e statistica
2020/21
The course aims at completing the students' education in basic mathematics, by introducing the theory of analytic functions, distributions, Fourier and Laplace transforms, and discrete and continuous probability. Such topics play an essential role in engineering applications. Examples and motivation will be drawn from problems in engineering, offering further insights.
The course aims at completing the students' education in basic mathematics, by introducing the theory of analytic functions, distributions, Fourier and Laplace transforms, and discrete and continuous probability. Such topics play an essential role in engineering applications. Examples and motivation will be drawn from problems in engineering, offering further insights.
a) Knowledge and understanding Students are taught some basic mathematical notions and tools for solving various problems ranging from signals analysis to the study of random phenomena. The theory of distributions provides a general language which enables to deal with signals arising in impulsive or discontinuous phenomena: this theory is the natural setting for the study of the Fourier and Laplace transforms. Students learn the techniques for the computation of the transforms of the main distributions: delta Dirac, Dirac comb, and piecewise regular functions included. Complex analysis is the proper setting for the theory of the Laplace transform and is the advanced tool for the analysis of singular phenomena and for the computation of integrals. Moreover, students are provided with the main probabilistic tools necessary for solving problems under uncertainty. They learn how to deal with random phenomena and with the variables involved in them. b) Applying knowledge and understanding At the end of the course students will be able to apply the analytical techniques required for the analysis of the signals of any nature (impulsive, erratic, etc.). Also, they will be expected to have acquired the skills to evaluate the probability of outcomes and extrapolate information useful in solving problems in electronic and telecommunication engineering. The ability to apply the gained knowledge will be verified through class exercises.
a) Knowledge and understanding Students are taught some basic mathematical notions and tools for solving various problems ranging from signals analysis to the study of random phenomena. The theory of distributions provides a general language which enables to deal with signals arising in impulsive or discontinuous phenomena: this theory is the natural setting for the study of the Fourier and Laplace transforms. Students learn the techniques for the computation of the transforms of the main distributions: delta Dirac, Dirac comb, and piecewise regular functions included. Complex analysis is the proper setting for the theory of the Laplace transform and is the advanced tool for the analysis of singular phenomena and for the computation of integrals. Moreover, students are provided with the main probabilistic tools necessary for solving problems under uncertainty. They learn how to deal with random phenomena and with the variables involved in them. b) Applying knowledge and understanding At the end of the course students will be able to apply the analytical techniques required for the analysis of the signals of any nature (impulsive, erratic, etc.). Also, they will be expected to have acquired the skills to evaluate the probability of outcomes and extrapolate information useful in solving problems in electronic and telecommunication engineering. The ability to apply the gained knowledge will be verified through class exercises.
Students are required to be familiar with the notions and tools of the mathematics courses of the first two years: these include differential and integral calculus of one or several variables.
Students are required to be familiar with the notions and tools of the mathematics courses of the first two years: these include differential and integral calculus of one or several variables.
1. (27h) Function theory of complex variable: differentiability, Cauchy-Riemann equations, line integrals. Cauchy theorem, Cauchy integral formula, Taylor series of analytic functions, Laurent series. Residue theorem, computation of residues and application to the calculation of integrals. 2. (15h) Theory of distributions: definitions and basic operations (algebraic operations, translation, rescaling, derivatives), Dirac delta, p.v.(1/t), Dirac comb. Convolution of functions and distributions. 3. (18h) Fourier and Laplace transforms of functions and tempered distributions: definitions and properties, inverse transforms, inversion formula. Notable transforms. 4. (15h) Combinatorics, probability measures and related elementary properties. Conditional probability and independence. 5. (15h) Discrete and continuous random variables. Notable examples. Expected values. 6. (10h) Joint distribution, independence and correlation.
1. (27h) Function theory of complex variable: differentiability, Cauchy-Riemann equations, line integrals. Cauchy theorem, Cauchy integral formula, Taylor series of analytic functions, Laurent series. Residue theorem, computation of residues and application to the calculation of integrals. 2. (15h) Theory of distributions: definitions and basic operations (algebraic operations, translation, rescaling, derivatives), Dirac delta, p.v.(1/t), Dirac comb. Convolution of functions and distributions. 3. (18h) Fourier and Laplace transforms of functions and tempered distributions: definitions and properties, inverse transforms, inversion formula. Notable transforms. 4. (15h) Combinatorics, probability measures and related elementary properties. Conditional probability and independence. 5. (15h) Discrete and continuous random variables. Notable examples. Expected values. 6. (10h) Joint distribution, independence and correlation. Exercises will cover the topics of the lectures. Some of them will be carried out by the teacher at the blackboard, others will actively involve the students.
Exercises will cover the topics of the lectures. Some of them will be carried out by the teacher, others will actively involve the students.
Lecture notes will be available in the course web page. Recommended textbook in probability: Ross, S. 'A first course in Probability', Pearson Ed., 2014 (or any other edition).
Lecture notes will be made available on the course web page. Recommended textbook in probability: Ross, S. 'A first course in Probability', Pearson Ed., 2014 (or any other edition).
Modalit� di esame: Prova orale facoltativa; Prova scritta tramite PC con l'utilizzo della piattaforma di ateneo;