ID:
500541
Duration (hours):
88
CFU:
9
SSD:
ANALISI MATEMATICA
Year:
2025
Overview
Date/time interval
Primo Semestre (29/09/2025 - 16/01/2026)
Syllabus
Course Objectives
Learn how to work in the complex framework, evaluate integrals of holomorphic functions, manipulate power and Fourier series, adopt the point of view of signal theory, calculate and operate with Fourier, Laplace and Zeta transforms, solve simple ordinary differential equations with constant coefficients, understand convolutions.
The second part (Optimization and discrete transforms, 3CFU only for Bioengineering) will be devoted to the elementary notions of free and constraint optimization and to the basic techniques of the mathematical theory of discrete signals (DFT, FFT, convolutions) with simple applications to difference equations and numerical approximations.
The second part (Optimization and discrete transforms, 3CFU only for Bioengineering) will be devoted to the elementary notions of free and constraint optimization and to the basic techniques of the mathematical theory of discrete signals (DFT, FFT, convolutions) with simple applications to difference equations and numerical approximations.
Course Prerequisites
Differential and integral calculus for scalar and vector functions, matrices and linear transformations, sequences and series, power series in the real line, complex numbers, polar coordinates.
Teaching Methods
The course is divided into lessons (on the blackboard, integrated with transparencies), exercises on the blackboard and laboratory activities.
During the lessons, the main results, their scope of validity, their mutual relationships, and the most relevant applications are presented and discussed.
Exercises and laboratory activities are aimed at acquiring the main calculation techniques and the most elaborate strategies for solving problems, in the context of the theoretical results already acquired. Part of the exercises is also aimed at solving the exam topics from previous years.
During the lessons, the main results, their scope of validity, their mutual relationships, and the most relevant applications are presented and discussed.
Exercises and laboratory activities are aimed at acquiring the main calculation techniques and the most elaborate strategies for solving problems, in the context of the theoretical results already acquired. Part of the exercises is also aimed at solving the exam topics from previous years.
Assessment Methods
The exam consists of a written test and, for the Discrete Transforms and Optimization module, a laboratory test.
The written test is divided into two parts. The first part consists in solving five problems, in which it is necessary to apply an articulated solution strategy, in any case discussed during the exercise hours.
The second part is of a more theoretical nature and concerns the knowledge of the main results presented in class and the understanding of the relationships between them. This second part consists of five problems as well. The overall grade is the average of the grades obtained in the individual parts; written tests in which each part has achieved a rating of less than 16 points are not considered acceptable.
The laboratory test verifies the learning of both the main concepts and the elaboration techniques, which were presented during the laboratory hours carried out during the course.
The written test is divided into two parts. The first part consists in solving five problems, in which it is necessary to apply an articulated solution strategy, in any case discussed during the exercise hours.
The second part is of a more theoretical nature and concerns the knowledge of the main results presented in class and the understanding of the relationships between them. This second part consists of five problems as well. The overall grade is the average of the grades obtained in the individual parts; written tests in which each part has achieved a rating of less than 16 points are not considered acceptable.
The laboratory test verifies the learning of both the main concepts and the elaboration techniques, which were presented during the laboratory hours carried out during the course.
Texts
G.C. Barozzi, Matematica per l'Ingegneria dell'informazione. Zanichelli 2004
F. Bagarello, Metodi matematici per fisici e ingegneri. Zanichelli 2019
M. Codegone, L. Lussardi.
Metodi Matematici per l'Ingegneria. Zanichelli 2021.
M. Giaquinta, G. Modica. Note di Metodi Matematici per Ingegneria Informatica. Pitagora, Bologna.
Lecture notes available on the web page of the course.
Nocedal, Jorge, Wright, S. Numerical Optimization. Springer Verlag, 2006.
F. Bagarello, Metodi matematici per fisici e ingegneri. Zanichelli 2019
M. Codegone, L. Lussardi.
Metodi Matematici per l'Ingegneria. Zanichelli 2021.
M. Giaquinta, G. Modica. Note di Metodi Matematici per Ingegneria Informatica. Pitagora, Bologna.
Lecture notes available on the web page of the course.
Nocedal, Jorge, Wright, S. Numerical Optimization. Springer Verlag, 2006.
Contents
Complex functions Manipulation of complex numbers Rational, exponential, and trigonometric functions, logarithms Power series Conplex derivatives, olomorphic functions, Cauchy-Riemann conditions Line integrals, Cauchy theorem, , analyticity of olomorphic functions Singularities, Laurent series, residue formula Evaluation of integrals, Jordan lemma The language of signals Continuous and discrete signals. Basic operations on signals: sum and linear combinations of signals, traslation and rescalings. Scalar products and norms. Z trasform Definition and simple examples Simple applications to difference equations Fourier series Periodic signals, trigonometric and exponential functions, Fourier series. Pointwise and energy convergence, Gibbs phenomenon. Parseval identity Applications Fourier Transform Definition of Fourier transform, relationships with Fourier series, elementary properties Riemann-Lebesgue lemma Inversion theorem for piecewise regular functions Plancherel identity, Fourier transform for L^2 functions Laplace transform Definition, links with the Fourier transform, main properties Inversion of Laplace transform, residue and Heaviside formula Application to simple ordinary differential equations Convolution Definition and simple example of convolutions Links with Fourier and Laplace transform Simple applications to differential equations Second part (3CFU): Unconstrained Optimization Problems - Gradient methods and line-searches - Newton and quasi-Newton methods (BFGS, DFP) - Trust region methods Discrete transforms Discrete Fourier transform (DFT) The algorithm of Fast Fourier Transform (FFT) Discrete convolution Applications to difference and approximation problems, stability
Course Language
Italian
More information
The course is divided in two parts. The first part is about the theory of Mathematical methods (6CFU). The second part, only for the Bioengineering program, Nonlinear Optimization and Discrete Transforms (3CFU).
Students in the categories identified by the project on innovative teaching will have the opportunity to hold receptions even at special times.
Students in the categories identified by the project on innovative teaching will have the opportunity to hold receptions even at special times.
Degrees
Degrees
Bioengineering
Bachelor’s Degree
3 years
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People
People (2)
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