## Learning objectives

Knowledge of series and transforms to apply to the theoretical and applied phisic.

## Prerequisites

Matematica 2

## Course unit content

Successions and series of functions. Complex Analysis. Fourier series. Fourier transform. Laplace transform.

## 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. Fourier ’s transform.

10. Fourier transform. Definition and properties.

11. Laplace transform. Definition and properties.

## Bibliography

Barozzi-Matarasso, "Metodi Matematici per l'Ingegneria", ed. Zanichelli.

Pagani-Salsa, "Analisi Matematica II" ed. Masson.

Spiegel, "Variabile Complessa", collana Schaum's.

## Teaching methods

Frontal lessons

## Assessment methods and criteria

The examination combines write text and oral discussion.

## Other information

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