First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
Course unit
SCO2045440, A.A. 2018/19

Information concerning the students who enrolled in A.Y. 2017/18

Information on the course unit
Degree course First cycle degree in
SC1158, Degree course structure A.Y. 2014/15, A.Y. 2018/19
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Number of ECTS credits allocated 6.0
Type of assessment Mark
Course unit English denomination FUNDAMENTS OF MATHEMATICAL METHODS
Website of the academic structure
Department of reference Department of Physics and Astronomy
Mandatory attendance No
Language of instruction Italian
Single Course unit The Course unit can be attended under the option Single Course unit attendance
Optional Course unit The Course unit can be chosen as Optional Course unit

Teacher in charge LUCA MARTUCCI FIS/02

ECTS: details
Type Scientific-Disciplinary Sector Credits allocated
Educational activities in elective or integrative disciplines FIS/02 Theoretical Physics, Mathematical Models and Methods 6.0

Course unit organization
Period Second semester
Year 2nd Year
Teaching method frontal

Type of hours Credits Teaching
Hours of
Individual study
Practice 2.0 24 26.0 No turn
Lecture 4.0 32 68.0 No turn

Start of activities 25/02/2019
End of activities 14/06/2019
Show course schedule 2019/20 Reg.2014 course timetable

Examination board
Board From To Members of the board
9 Istituzioni di Metodi Matematici (iniziali cognome M-Z) 01/10/2018 30/11/2019 VOLPATO ROBERTO (Presidente)
MARTUCCI LUCA (Membro Effettivo)
MATONE MARCO (Supplente)
8 Istituzioni di Metodi Matematici (iniziali cognome A-L) 01/10/2018 30/11/2019 MARTUCCI LUCA (Presidente)
VOLPATO ROBERTO (Membro Effettivo)
MATONE MARCO (Supplente)
7 Istituzioni di Metodi Matematici (iniziali cognome M-Z) 01/10/2017 30/11/2018 VOLPATO ROBERTO (Presidente)
MARTUCCI LUCA (Membro Effettivo)
MATONE MARCO (Supplente)
6 Istituzioni di Metodi Matematici (iniziali cognome A-L) 01/10/2017 30/11/2018 MARTUCCI LUCA (Presidente)
VOLPATO ROBERTO (Membro Effettivo)
MATONE MARCO (Supplente)

Prerequisites: Appropriate knowledge of the content of the courses "Analisi matematica I,II,III", including Lebesgue measure theory.
Target skills and knowledge: Complex analysis, Hilbert spaces and distributions.
Examination methods: The exam is divided in two parts: resolution of exercises and theory
Assessment criteria: The student is expected to demonstrate both knowledge of the theory and skills in solving exercises.
Course unit contents: A. Analytic functions

1. Cauchy-Riemann conditions.

2. Laplacian on C. Harmonic and analytic functions. Determination of an analytic function from its real or imaginary part.

3. Conformal transformations and analytic functions.

4. Integration over C. Darboux inequality. Cauchy theorem. Fundamental theorem of calculus, Morera theorem. Cauchy formula. Average theorem, maximum modulus principle, Liouville theorem, fundamental theorem of algebra.

5. Complex series. Weierstrass theorem. Power series, Abel theorem, Taylor series theorem, Laurent series.

6. Isolated singularities (removable, poles, essential). Picard theorems on essential singularities. Residues. Point at infinity. Multi-valued functions and branch points.

7. Zeroes, uniqueness theorems.

8. Residue theorem. Residue at infinity.

9. Index theorem for zeroes and poles. Argument principle.

10. Residue theory applied to integral calculus, Jordan Lemma and its applications. Integrals of trigonometric functions.

11. Principal part of an integral. Epsilon prescription.

12. Integrals involving multi-valued functions

B. Hilbert spaces and distributions

1. Finite and infinite dimensional vector spaces. Dirac notation. Pre-Hilbert and normed spaces.

2. Convergence, completeness, completion theorem. Banach and Hilbert spaces. Important examples: spaces l_2 and L_2.

3. Subspaces. Orthogonal complement. Decomposition in orthogonal subspaces.

4. Orthonormal systems and basis (o.n.s. & o.n.b.). Gram-Schmidt procedure. Separability and numerability of o.n.s. Fourier expansion in o.n.b. Riesz-Fischer theorem. Examples of o.n.b.
(Legendre, Hermite and Leguerre polynomials)

5. Continuous and bounded linear functionals, Riesz theorem, Dirac notation.

6. Schwarz spaces, tempered distributions, operations on distributions.

6. Riesz Lemma.

7. Bounded linear operators: adjoint e inverse operator, analytic function of an operator, self-adjoint operators, orthogonal projectors

8. Fourier transform and its extension to distributions. FT and convolution. FT as unitary transformation on L_2.

9. Adjoint of unbounded operators. Symmetric, self-adjoint and essentially self-adjoint operators. Important examples: operators X, P, P^2
Planned learning activities and teaching methods: Blackboard lectures
Textbooks (and optional supplementary readings)
  • Smirnov, Corso di Matematica Superiore, vol. 3 parte II. --: Ed. Riuniti, --. Cerca nel catalogo
  • Rossetto, Metodi Matematici della Fisica. --: Ed. Levrotto e Bella, --. Cerca nel catalogo
  • Musso e Ragnisco, Raccolta di Esercizi e Problemi di Analisi Complessa e Algebra Lineare. --: Aracne, --. Cerca nel catalogo
  • Pradisi, Lezioni di Metodi Matematici per la Fisica. --: Ed. della Normale, --. Cerca nel catalogo
  • Onofri, Lezioni sulla Teoria degli Operatori Lineari. --: Ed. Zara, --. Cerca nel catalogo
  • Abbati e Cirelli, Metodi Matematici per la Fisica. Operatori Lineari negli Spazi di Hilbert. --: Ed. Città Studi, --. Cerca nel catalogo
  • Kolmogorov e Fomin, Elementi della Teoria delle Funzioni e di Analisi Funzionale. --: Ed. Riuniti, --. Cerca nel catalogo
  • Weidmann, Linear Operators in Hilbert Spaces. --: Ed. Springer-Verlag, --. Cerca nel catalogo

Innovative teaching methods: Teaching and learning strategies
  • Lecturing
  • Loading of files and pages (web pages, Moodle, ...)

Innovative teaching methods: Software or applications used
  • Moodle (files, quizzes, workshops, ...)