## Learning objectives

To provide basic intruments for the calculus with the complex variable, the functions series and the Fourier and Laplace transforms.

## Prerequisites

Knowledge of the properties of the real functions.

## Course unit content

Function series. Complex variable. Fourier and Lapklace transforms.

## Full programme

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.

Liouville’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.

## Bibliography

G.C. Barozzi , Matematica per l' Ingegneria dell' Informazione, ed. Zanichelli.

M.R. Spiegel , Variabili Complesse , collana Schaum's , Mc Graw-Hill.

## Teaching methods

Frontal lessons followed by learning tests.

## Assessment methods and criteria

Written tests followed by oral tests.

## Other information

Anyone