


Politecnico di Torino  
Academic Year 2010/11  
05BQXNX Mathematical methods for engineers 

1st degree and Bachelorlevel of the Bologna process in Electronic Engineering  Torino 





Subject fundamentals
The purpose of this course is to complete the students' formation in basic mathematics, introducing the theory of analytic functions, distributions, Fourier and Laplace transforms, and discrete and continuous probability. Such topics have an important role in engineering applications; during the course many examples illustrating these applications and offering further insight will be discussed.

Expected learning outcomes
The knowledge obtained through this course will consist of some basic mathematical concepts and tools commonly used to solve different problems, varying from signal analysis to nondeterministic phenomena. Distribution theory offers the student a general and flexible language to deal with signals of various type (impulsive, discontinuous, etc.) and is also the natural context in which to study Fourier and Laplace transforms. The student will learn efficient techniques with which to compute such transforms including a number of fundamental examples (transforms of delta's, trains of delta's, discontinuous functions). Complex function theory offers a language to adequately analyze the Laplace transform, and it offers advanced tools to study singular phenomena and to compute integrals. The probabilistic tools acquired are those commonly used to solve simple nondeterministic problems, characterized by the unpredictable behavior of items or the lifetime of systems. By the end, students will be able to evaluate the probability of outcomes and extrapolate information useful in planning problems in electronic and telecommunication engineering. The ability to apply the gained knowledge will be verified through class exercises.

Contents
Complex variable functions: derivability, CauchyRiemann conditions, line integrals. Cauchy theorem, residue theorem, integral Cauchy formula, residue calculus, computation of integrals using residues. Power and Laurent series expansions of analytic functions.
Distribution theory: definition and basic operations (algebraic operations, translation, scaling, derivation). Some important distributions: delta, train of deltas, p.v. 1/t. Convolution of functions and distributions. Fourier and Laplace transform for functions and tempered distribution: definition, properties, the inverse transform. Some important examples Elements of combinatorial calculus, probability spaces and elementary properties of probability measures. Discrete and continuous random variables, distribution of a random variable. Some notable examples. Mean values, joint distribution, independence and correlation, conditional mean. 
Delivery modes
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.

Texts, readings, handouts and other learning resources
' Ross, S. 'Calcolo delle Probabilità', Ed. APOGEO, 2007.
' Baldi, P. 'Introduzione alla probabilità, con elementi di statistica', McGrawHill, 2007 
Assessment and grading criteria
Written examination

