The course aims to provide the basics of functions of one complex variable, Fourier analysis, functional transforms, as well as to introduce students to solving problems of practical interest which exploit these mathematical tools.
D1 - Knowledge and understanding skills
At the end of the course, the student should know the fundamentals of functions of one complex variable, Fourier analysis, functional transforms.
D2 - Applying knowledge and understanding
The student should be able to deal with and solve simple exercises, questions, problems, of theoretical and computational issues, related to the topics of the course.
D3 - Making judgements
The student should be able to describe, model and solve simple problems of interest for applications, by using the mathematical tools developed during the course.
D4 - Communication skills
The student should be able to describe mathematical topics with an adequate command of language, as well as to translate practical problems in mathematical terms.
D5 - Learning skills
The student should be able to read and understand books and articles using the mathematical tools learned in the course, and be able to learn more advanced ones.

Complex numbers.
Linear algebra. Differential and integral calculus for function of several variables. Numerical and function series. Ordinary differential equations.

The set C of complex numbers: the algebraic structure and the metric structure, cartesian form and polar form of a complex number, De Moivre formulas.

Complex functions with complex variable: continuous, differentiable, holomorfphic functions, the monogeneity conditions; power series, exponential functions, circular functions, hyperbolic functions and their properties; the nth root and the logarithm.

Elements of complex analysis: parametric curves in C, integral on a curve of a complex function, Cauchy's Theorem and Cauchy's integral formulas for a function; principal theorems of complex analysis; analytic functions; properties of the zeros of an analytic function; classification of the singularities of a function and residue of a function in an isolated singular point; bilateral series and Laurent theorem; the theorem of the residues; calculation of integrals with the residue method.

Introduction to the Lebesgue integral: definition of the Lebesgue integral, dominated convergence theorem, Fubini-Tonelli theorem, parameter-dependent integrals.

Elements of functional analysis: Banach and Hilbert spaces, orthogonal systems, best approximation theorem, Lebesgue spaces and properties, convolution and properties.

Fourier series and applications: Fourier polynomials and series, Bessel inequality and Parseval identity, pointwise, uniform and L^2-convergence of Fourier series, term by term differentiation and integration, regularity of a function and speed of decay of its Fourier coefficients, applications of Fourier series to the solution of the heat and the wave equation.

Fourier transform and applications: Fourier transform in L^1 and its properties, calculation of Fourier transforms, approximating kernels and approximated identities, inverse Fourier transform, Fourier inversion theorem in L^1, Fourier transform in L^2, Plancherel theorem, application of the Fourier transform to sampling problems, Shannon theorem, applications of the Fourier transform to the solution of some classes of differential equations.

Laplace transforms: definition and main properties; the problem of the reverse transformation; application of the transforms to some classes of differential, integral and integro-differential equations and systems.

Notes on the distributions.

- G.C. Barozzi, Matematica per l’ingegneria dell’informazione, Zanichelli, Bologna, 2001.
- G. Gilardi, Analisi tre, McGraw-Hill, Milano, 2003.
- M. Codegone, Metodi matematici per l’Ingegneria, Zanichelli, Bologna, 1995.

The set C of complex numbers: the algebraic structure and the metric structure, cartesian form and polar form of a complex number, De Moivre formulas.
Complex functions with complex variable: continuous, differentiable, holomorfphic functions, the monogeneity conditions; power series, exponential functions, circular functions, hyperbolic functions and their properties; the nth root and the logarithm.

Elements of complex analysis: parametric curves in C, integral on a curve of a complex function, Cauchy's Theorem and Cauchy's integral formulas for a function; principal theorems of complex analysis; analytic functions; properties of the zeros of an analytic function; classification of the singularities of a function and residue of a function in an isolated singular point; bilateral series and Laurent theorem; the theorem of the residues; calculation of integrals with the residue method.

Introduction to the Lebesgue integral: definition of the Lebesgue integral, dominated convergence theorem, Fubini-Tonelli theorem, parameter-dependent integrals.

Elements of functional analysis: Banach and Hilbert spaces, orthogonal systems, best approximation theorem, Lebesgue spaces and properties, convolution and properties.

Fourier series and applications: Fourier polynomials and series, Bessel inequality and Parseval identity, pointwise, uniform and L^2-convergence of Fourier series, term by term differentiation and integration, regularity of a function and speed of decay of its Fourier coefficients, applications of Fourier series to the solution of the heat and the wave equation.

Fourier transform and applications: Fourier transform in L^1 and its properties, calculation of Fourier transforms, approximating kernels and approximated identities, inverse Fourier transform, Fourier inversion theorem in L^1, Fourier transform in L^2, Plancherel theorem, application of the Fourier transform to sampling problems, Shannon theorem, applications of the Fourier transform to the solution of some classes of differential equations.

Laplace transforms: definition and main properties; the problem of the reverse transformation; application of the transforms to some classes of differential, integral and integro-differential equations and systems.

Notes on the distributions.

The detailed program is available at:
http://www.dmi.units.it/~obersnel/#insegnamenti and in Moodle http://moodle2.units.it;

Lectures and classroom exercises. Homework available on the web site.

See http://www.dmi.units.it/~obersnel/
Some information and teaching materials are available on the Moodle site of the course.

Written and oral examinations, concerning the practical (exercises) and the theoretical aspects (definitions, theorems, proofs) of the course. The oral exam can be accessed only after passing the written test.