MATHEMATICAL METHODS
Stampa
Enrollment year
2020/2021
2021/2022
Regulations
DM270
MAT/05 (MATHEMATICAL ANALYSIS)
Department
DEPARTMENT OF ELECTRICAL,COMPUTER AND BIOMEDICAL ENGINEERING
Course
ELECTRONIC AND COMPUTER ENGINEERING
Curriculum
PERCORSO COMUNE
Year of study
Period
1st semester (27/09/2021 - 21/01/2022)
ECTS
6
Lesson hours
60 lesson hours
Language
Italian
Activity type
WRITTEN AND ORAL TEST
Teacher
GIANAZZA UGO PIETRO (titolare) - 6 ECTS
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.
Learning outcomes
Learn how to work with complex variable functions, 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.
Course contents
Complex variable functions.
Manipulation of complex numbers.
Rational, exponential, and trigonometric functions, logarithms.
Power series.
Complex derivatives, holomorphic functions, Cauchy-Riemann conditions.
Contour 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 transform.
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
Teaching methods
The course is divided into lessons (on the blackboard, possibly supplemented by transparencies) and exercises on the blackboard.
During the lessons the main results, their scope of validity, the mutual relations, and the most relevant applications are presented and discussed.
The exercises 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 addressed to the solution of the written exams of the previous years.