Lecturer: Ugo Pietro Gianazza  

Course name: Mathematical methods
Course code: 500541
Degree course: Ingegneria Elettronica e delle Telecomunicazioni, Ingegneria Industriale
Disciplinary field of science: MAT/05
L'insegnamento costituisce attività di base per: Ingegneria per l'Ambiente e il Territorio, Ingegneria Industriale
University credits: CFU 6
Course website: http://www.imati.cnr.it/~gianazza/metodi.html

Specific course objectives

Learn how to work in the complex framework, evaluate integrals of olomorphic functions, manipulate power and Fourier series, adopt the point of view of signal theory, calculate and operate with Fourier and Laplace transforms, solve simple ordinary differential equations with constant coefficients, understand convolutions.

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.

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

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

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): 22
Practical class (hours/year in lecture theatre): 43
Practicals / Workshops (hours/year in lecture theatre): 0

Suggested reading materials

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

M. Giaquinta, G. Modica. Note di Metodi Matematici per Ingegneria Informatica. Pitagora, Bologna.

F. Tomarelli. 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.