COMPLEMENTI DI ANALISI MATEMATICA

1. Periodic, piecewise continuous and regular functions. Developments in Fourier series. Pointwise and uniform convergence of Fourier series, integration term by term. Calculating the sums of convergent numerical series.

2. Derivation and integration in the complex field. Cauchy formula, Liouville theorem, proof of the fundamental theorem of algebra. Theorem of Hermite. Laurent theorem on the developability in two-sided power series. Isolated singular points, classification and characterization. Calculation of residues in the poles, the residue theorem and its applications.

3. Fourier transformation. Definition and basic properties. Transforms of the functions rect (x), exp (-ax^2) and exp (-a | x |) with a> 0, 1 / (1 + x^2). Derivative and transforms. Convolutions and their transforms. Inversion formulas.

4. Laplace transformation. Definition and basic properties. Transforms the functions H (t), sin (ωt), cos (ωt), [t]. Transform of periodic functions. Derivative and transformed, the final value theorem. Convolutions and their transformed. inversion formula. Applications to linear differential equations and systems with constant coefficients.

5. Outline of distributions. The test function space. Distributions. The L^1_loc (R) space. Distribution functions. The Dirac distribution. Sequences of distributions. Operations. Derivative of a distribution. Significant special cases.

1) N. FUSCO - P. MARCELLINI - C. SBORDONE, Elementi di Analisi Matematica due, Liguori, Napoli, 2001.

2) G. DI FAZIO - M. FRASCA, Metodi Matematici per l’Ingegneria, Monduzzi, Bologna, 2003.

3) G. C. BAROZZI, Matematica per l’Ingegneria dell’Informazione, Zanichelli, Bologna, 2003.