Lecturer: Ugo Pietro Gianazza  

Course name: Mathematical methods
Course code: 500541
Degree course: Ingegneria Meccatronica
Disciplinary field of science: MAT/05
University credits: CFU 6
Course website: http://www.imati.cnr.it/~gianazza/metodi.html

Specific course objectives

The course aims to introduce some of the main methods of mathematical analysis, and to provide students with useful operative tools for applications to signal theory and optimization problems. Main objectives are: i) lead the students to easily use the most important functions of complex variable and provide the basic knowledge of the related theory, ii) introduce the concept of convergence of sequences and series of functions and present the fundamental results about Fourier series, Fourier transform and Laplace transform iii) illustrate some techniques and applications of these transformations to simple differential problems.

Course programme

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.

Introduction to complex analysis

• Complex numbers
•  Power series in complex fields: radius of convergence and formula for its calculation
•  Exponential function, trigonometric functions, roots and logarithm of complex numbers
•  Complex derivative and holomorphic functions, holomorphism of power series
•  Path integrals in the complex field
•  Cauchy theorem, analyticity of holomorphic functions
•  Singularity and Laurent series, residue Theorem
•  Application to the calculation of integrals, Jordan Lemma.

Fourier series

• Periodic signals, trigonometric polynomials, Fourier series, comparison between trigonometric and exponential forms
•  The best approximation problem and convergence in the energy sense
•  Parseval identity and application to the calculation of numeric series
• Pointwise and uniform convergence, application to the calculation of numeric series

Fourier transform for integrable functions

• Definition of Fourier transform, fundamental properties, link with Fourier series
• Riemann-Lebesgue lemma, examples of calculation of Fourier transforms
• Fourier transform of signal with finite energy and the Plancherel Theorem
• The inversion theorem

Laplace transform

• Definition, main properties and examples of calculation
• Link with Fourier transform
• Inversion of Laplace transform, Heaviside formula.
• Application to linear differential equations with constant coefficients and to the Cauchy problem

Convolution

• Definition and main properties, examples
• Link with Fourier and Laplace transformation

Z trasform

• Definition and simple examples
• Simple applications to difference equations

Course entry requirements

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.

Course structure and teaching

Lectures (hours/year in lecture theatre): 38
Practical class (hours/year in lecture theatre): 14
Practicals / Workshops (hours/year in lecture theatre): 0

Suggested reading materials

M. Codegone. Metodi Matematici per l'Ingegneria. Zanichelli.

F. Tomarelli. Esercizi di Metodi Matematici per l'Ingegneria. CLU.

Testing and exams

A written and an oral examination, the latter one conditioned by the outcome of the former one. Both examinations must taken in the same exam session.