Learning outcomes of the course unit
To provide basic intruments for the calculus with the complex variable, the functions series and the Fourier and Laplace transforms.
Knowledge of the properties of the real functions.
Course contents summary
Function series. Complex variable. Fourier and Lapklace transforms.
In the following we will intend n.p=no proof.
1. Successions of functions. Punctual convergence. Uniform convergence. Cauchy’s criterion. Theorem of
boundedness. Theorem of change of limits (n.p.). Theorem of continuity. Theorem of integrability (n.p.).
Theorem of derivability.
2. Series of functions. Punctual, uniform and absolute convergence. Cauchy’s criterion. Cauchy’s N.C. Total
convergence. Weierstrass’ criterion. Theorems of boundedness, continuity, integradility, derivability.
3. Complex numbers. Cartesian, polar and exponential forms. Complex functions.
4. Holomorphic functions. Complex derivative. Cauchy-Riemann conditions. Confront with the real
differentiability. De l’ Hopital’s theorem (n.p.).
5. Power series. Radius of convergence. Term by term derivability. Abel’s criterion. Taylor’s series. Expansion
of elementary functions.
6.Fourier series. Punctual convergence. Uniform convergence. Quadratic mean convergence. Bessel’s inequality.
Parseval’s identity. Fischer-Riesz theorem.
7. Countour integrals. Cauchy’s theorem. Cauchy’s integral representation formula. Mean value theorem.
Maximum principle’s theorem. Fundamental theorem of Algebra. Existence of a primitive.. Morera’s theorem.
8. Laurent’s series. Isulated singularities : classification and characterization. Isulated singularity at infinity.
Residue in a point and at infinite. The Cauchy’s residues theorem.
9. Principle value of improper integrals. Great circle lemma. Jordan’s lemma.
10. Fourier Transform (FT) of summable functions of one real variable. Definition, properties and examples.
11. Laplace Transform (LT).Definition, properties and examples.
G.C. Barozzi , Matematica per l' Ingegneria dell' Informazione, ed. Zanichelli.
M.R. Spiegel , Variabili Complesse , collana Schaum's , Mc Graw-Hill.
Frontal lessons followed by learning tests.
Assessment methods and criteria
Written tests followed by oral tests.