PHYSICS | University of Parma

Professor: Silvana MARCHI
Academic discipline: Analisi matematica (MAT/05)
Field: Discipline matematiche e informatiche
Type of training activity: Basic

48 hours
of face-to-face activities

6 credits

hub:

course unit
in - - -

lectures timetable unavailable

exam calls

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

2030 agenda goals for sustainable development

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Contacts

Toll-free number

800 904 084

Student registry office

Tel. +39 0521 905116
E-mail segsmfn@unipr

Quality assurance office

Education manager:
Dr. Marco Squarcia
Tel. +39 0521 906094
Office E-mail segreteria.smfi@unipr
Manager E-mail marco.squarcia@unipr.it

President of the degree course

Prof. Luigi Cristofolini
E-mail luigi.cristofolini@unipr.it

Deputy President of the degree course

Prof.ssa Eugenia Polverini
E-mail eugenia.polverini@unipr.it

Faculty advisor

Prof. Danilo Bersani
E-mail danilo.bersani@unipr.it

Prof.ssa Antonella Parisini
E-mail: antonella.parisini@unipr.it

Prof. Francesco Cugini
E-mail: francesco.cugini@unipr.it

Career guidance delegate

Prof. Alessio Bosio
E-mail alessio.bosio@unipr.it

Tutor Professors

Prof. Stefania Abbruzzetti
E-mail stefania.abbruzzetti@unipr.it

Prof. Andrea Baraldi
E-mail andrea.baraldi@unipr.it

Prof. Francesco Di Renzo
E-mail francesco.direnzo@unipr.it

Prof. Massimo Solzi
E-mail massimo.solzi@unipr.it

Erasmus delegates

Prof. Bersani Danilo
E-mail: bersani.danilo@unipr.it

Prof. Guido D'Amico
E-mail:guido.damico@unipr.it

Quality assurance manager