ID:
501997
Duration (hours):
48
CFU:
6
SSD:
FISICA TEORICA, MODELLI E METODI MATEMATICI
Year:
2025
Overview
Date/time interval
Secondo Semestre (02/03/2026 - 05/06/2026)
Syllabus
Course Objectives
The main objective of the course is to present and familiarize students with the structures, basic mathematical models and calculation techniques necessary for the formulation and application of modern physics.
At the end of the course the students must have acquired the following skills and abilities:
- Knowing how to use the deductive and inductive method to prove theorems
- Correct use of formalism and basic mathematics tools for the development of knowledge of classical and quantum physics
- Understanding and knowing how to use the following mathematical structures with related calculation techniques: Finite and infinite vector spaces on the complex field, functions of complex variables and integration into the complex plane.
- Knowing how to organize, structure and expose the arguments carried out independently
- Knowing how to structure and design the strategy for solving mathematical problems involving the use of the mathematical structures listed above
- Understand the use of the mathematical structures previously listed in some simple classical physics problems
- Knowing how to evaluate the consistency of the results obtained
At the end of the course the students must have acquired the following skills and abilities:
- Knowing how to use the deductive and inductive method to prove theorems
- Correct use of formalism and basic mathematics tools for the development of knowledge of classical and quantum physics
- Understanding and knowing how to use the following mathematical structures with related calculation techniques: Finite and infinite vector spaces on the complex field, functions of complex variables and integration into the complex plane.
- Knowing how to organize, structure and expose the arguments carried out independently
- Knowing how to structure and design the strategy for solving mathematical problems involving the use of the mathematical structures listed above
- Understand the use of the mathematical structures previously listed in some simple classical physics problems
- Knowing how to evaluate the consistency of the results obtained
Course Prerequisites
Courses of Algebra and analysis in the first two years.
Teaching Methods
Aim of the course is to provide the basic knowledge of complex analysis. Attention is paid to the applications, mainly consisting in the solutions of integrals in the complex plane, with the aim to provide the necessary tools to deal with calculus techniques applied in physics.
The course is integrated with practice lectures, which are scheduled in the timetable of the course.
The course is integrated with practice lectures, which are scheduled in the timetable of the course.
Assessment Methods
Written and oral exams. The written test has a maximum duration of 3 hours and consists in the solutions of 3 exercises, concerning properties of the complex functions, series expansions, and solutions of integrals in the complex plane. The threshold of the written exam for the admission to the oral exam is 18/30. The oral exam consists of two questions on theorems of complex analysis and a question on the properties of Hilbert spaces.
The written and oral exams aim to verify the competence in solving problems and exercises on the topics presented in the course and the ability to dicuss in a mindful way the theoretical contents explained during the lectures.
The written and oral exams aim to verify the competence in solving problems and exercises on the topics presented in the course and the ability to dicuss in a mindful way the theoretical contents explained during the lectures.
Texts
1) Lecture notes written by the lecturer
2) Churchill, R.V., Brown, J.W., and Verhey,R.F.: Complex Variables and Applications, third edition, McGraw Hill (1976)
3) Jerrold E. Marsden, Michael J. Hoffman, "Basic Complex Analysis", W H Freeman & Co, (25 gennaio 1999)
2) Churchill, R.V., Brown, J.W., and Verhey,R.F.: Complex Variables and Applications, third edition, McGraw Hill (1976)
3) Jerrold E. Marsden, Michael J. Hoffman, "Basic Complex Analysis", W H Freeman & Co, (25 gennaio 1999)
Contents
2) Summary of the definition and properties of complex numbers. Definition and basic properties of analytical functions in the complex plane - Contour integrals of complex functions and Cauchy theorems - Cauchy's integral formula and inifinite differentiability of analytical functions - Taylor Series - Laurent Series - Isolated singularities - Residues Theorem -Multivalued functions and Riemann surfaces -
Applications of residues theorem to the calculations of generalized integrals
2) Normed spaces and Banach's spaces - Strong convergence in a normed space - Prehilbert and Hilbert spaces - Orthonormal systems and complete orthornormal sytems - Schwarz and Bessel inequalities - Generalized Fourier series and Parssval's identity - Gram-Schmidt construction - Isomorphism of Hilbert spaces - Projection theorem - Operators and linear functions in Hilbert spaces - Riesz-Fréchet theorem - Weak convergence and weak completeness of Hilbert spaces.
Applications of residues theorem to the calculations of generalized integrals
2) Normed spaces and Banach's spaces - Strong convergence in a normed space - Prehilbert and Hilbert spaces - Orthonormal systems and complete orthornormal sytems - Schwarz and Bessel inequalities - Generalized Fourier series and Parssval's identity - Gram-Schmidt construction - Isomorphism of Hilbert spaces - Projection theorem - Operators and linear functions in Hilbert spaces - Riesz-Fréchet theorem - Weak convergence and weak completeness of Hilbert spaces.
Course Language
Italian
More information
Students who fall under special categories (refer to portale.unipv.it/it/didattica/servizi-lo-studente/modalita-didattiche-inclusive for details) have access to course materials via KIRO. Upon request, they may also be granted permission to view recorded lectures from previous academic years through a dedicated link activated on KIRO. They are invited to contact the lecturer for online meetings and eventual group activities.
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PHYSICS
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