MATHEMATICAL MODELS AND METHODS OF PHYSICS

Course objectives

PART I General goals: The goal of the course is the study and the comprehension of advanced mathematical techniques. The student will acquire the basic concepts of complex analysis and functional analysis and she/he will be able to apply them to the study of problems in classical (electromagnetism and continuum media) and quantum physics. The know-how that the student acquires attending this course is indispensable for the Physics courses of the third year. Specific goals: Introduction to the fundamental concepts of complex analysis, i) to provide the student with a deep knowledge and understanding of these concepts, and ii) to allow him/her to successfully apply them in various physical contexts. In particular, the student must be able to use techniques of integration in the complex domain in all the physical contexts in which they have applications. In order to achieve these goals, and to help the student to develop the capability i) to communicate the acquired knowledges, and ii) to continue the studies autonomously, we plan to involve him (her), during the theoretical lectures and exercises, through general and specific questions related to the subject; or through the presentation in depth of some specific subject agreed with the teacher PART II General goals: The goal of the course is the study and the comprehension of advanced mathematical techniques. The student will acquire the basic concepts of complex analysis and functional analysis and she/he will be able to apply them to the study of problems in classical (electromagnetism and continuum media) and quantum physics. The know-how that the student acquires attending this course is indispensable for the Physics courses of the third year. Specific goals: Introduction to the fundamental concepts of functional analysis, i) to provide the student with a deep knowledge and understanding of these concepts, and ii) to allow him/her to successfully apply them in various physical contexts. In particular, the student must be able to use the Fourier series, the Fourier and Laplace transforms, and the distributions in all the physical contexts in which they have applications; in addition the student must be able to work with Hilbert spaces and with operators on functional spaces of physical interest. In order to achieve these goals, and to help the student to develop the capability i) to communicate the acquired knowledges, and ii) to continue the studies autonomously, we plan to involve him (her), during the theoretical lectures and exercises, through general and specific questions related to the subject; or through the presentation in depth of some specific subject agreed with the teacher

Channel 1
ALFREDO LEONARDO URBANO Lecturers' profile

Program - Frequency - Exams

Course program
Complex numbers and their properties. Analytic functions. Multivalued functions. Complex integrals. Cauchy theorem and Cauchy integral formula. Liouville and Morera theorems. The fundamental theorem of algebra. Maximum modulus theorem. Singularities and their classification. Taylor and Laurent series. Residues theorem and applications. Banach spaces. Hilbert spaces. Linear functionals and distributions. Linear operators in Hilbert spaces, self-adjoint operators, unitary operators, and spectrum. Lp spaces. Fourier series. Orthogonal polynomial sequences. Fourier and Laplace transform. Applications: linear ordinary and partial differential equations relevant in physics. Green's function.
Prerequisites
The fundamental prerequisite is the basic knowledge of the courses in mathematics of the first year and of the first semester of the second year. In particular, specific knowledge in the following fields is required: 1. Basic concepts of Calculus: real analysis of functions of one or many variables; derivatives, integrals, series. 2. Basic concepts of Linear Algebra. It is important that the student has knowledge of classical Physics, in particular mechanics and thermodynamics.
Books
C. Bernardini, O. Ragnisco, P. M. Santini "Metodi Matematici della Fisica", Carocci, 2014. M. W. Hirsch, S. Smale and R. L. Devaney, "Differential Equations, Dynamical Systems, and an Introduction to Chaos", Academic Press, 2012. M. Petrini, G. Pradisi, A. Zaffaroni, "A Guide to Mathematical Methods for Physicists", World Scientific. F. Calogero, "Metodi Matematici della Fisica", Dispense Istituto di Fisica, Universita' di Roma, 1975. F. Cesi "Rudimenti di analisi infinito dimensionale", dispense. N. Kolmogorov, S. V. Fomin, "Elementi di teoria delle funzioni e di analisi funzionale", Editori Riuniti.
Teaching mode
The format of the course consists of lectures at the blackboard.
Frequency
The course is based on the in-class frequency of traditional chalkboard lectures.
Exam mode
The final grading will be based on a written and an oral exams. The written exam consists of a test on the topics covered during the course. To pass the written test, the student should be able to carry out exercises on arguments explained during the course and apply the method that he/she learned to examples similar to the one discussed. For the evaluation the following points will be considered: - accuracy of the concepts laid out - clarity and accuracy of the exposition - ability to analytically develop the theory - ability in problem-solving (method and results) The oral exam consists of a discussion on the topics covered during the course. To pass the oral exam the student should be able to present an argument or repeat a calculation discussed during the course and apply the method that he/she learned to examples similar to the one discussed. For the evaluation the following points will be considered: - accuracy of the concepts laid out - clarity and accuracy of the exposition - ability to analytically develop the theory
Bibliography
C. Bernardini, O. Ragnisco, P. M. Santini "Metodi Matematici della Fisica", Carocci, 2014. M. W. Hirsch, S. Smale and R. L. Devaney, "Differential Equations, Dynamical Systems, and an Introduction to Chaos", Academic Press, 2012. L. V. Ahlfors, "Complex Analysis", Mc Graw-Hill 1979. M. Petrini, G. Pradisi, A. Zaffaroni, "A Guide to Mathematical Methods for Physicists", World Scientific. C. Presilla, "Elementi di Analisi Complessa" (2a edizione), Springer, UNITEXT 2014. F. Calogero, "Metodi Matematici della Fisica", Dispense Istituto di Fisica, Universita' di Roma, 1975. F. Cesi "Rudimenti di analisi infinito dimensionale", dispense.
Lesson mode
The format of the course consists of lectures at the blackboard.
ANDREA CAPUTO Lecturers' profile
Channel 2
VIERI MASTROPIETRO Lecturers' profile
Channel 3
LORENZO CAPRINI Lecturers' profile

Program - Frequency - Exams

Course program
1) Complex numbers introduction. * Operatios and geometric representation of complex numbers 2) Function of complex numbers and theory of analytic functions. * Introduction to functions of complex variables e differentiability. * Definition and properties of analytic functions. * polydrome functions and branch point. * Relation between causality and analytic functions. 3) Integration of complex functions. * Integrals in the complex plane. * Taylor and Laurent series and residue calculation. * Residue method to solve integrals of complex functions. * Residue method to solve integrals of real functions. * Principal value integrals. 4) Asymptotic expansions. * Laplace method. * Stationary phase method. * Steepest descent method.
Prerequisites
It is mandatory to know linear algebra in finite dimensional vector spaces. In particular, R, R2 and R3. It is mandatory to know total and partial derivatives, as well as integration of functions of one or more variables. It is important to have good understanding of the fundamental theorems on the convergence of series and integrals. It is important to have good understanding on how to perform a change of basis, how to diagonalize and how to solve an eigensystem, at least for the simplest cases of R2 and R3.
Books
Ablowitz, Fokas "Complex Variables: introduction and applications", Cambridge University Press. Presilla, "Elementi di Analisi Complessa: Funzioni di una variabile", Springer. Zanghì, "Appunti di Metodi Matematici della Fisica", Università di Genova, downloadable from: https://www.ge.infn.it/~zanghi/metodi/ZUL.pdf. Calogero, "Metodi Matematici della Fisica", Sapienza, lecture notes available online. Bernardini, Ragnisco, Santini, "Metodi Matematici della Fisica", Carocci Editorie.
Frequency
Optional
Exam mode
During the written exam it is allowed to use only one book. Formula sheets and personal notes can be used upon request to the teacher and only under their supervision. To access the oral examination it is necessary to have passed the written one with a grade equal or higher than 18. A passing grade in the written exam allows to take one oral examination only. If the final grade is not passing or it is rejected by the student, the grade of the written part is lost. Only the last submitted written exam is valid, all previous results are deleted. Students can decide not to take the oral examination, in which case, the final grade is the lowest between the result of the written part and 25. Students can retake the exam during the next appeal of the same session. The grade of the written exam remains valid until the Winter session 2024. Hence, the grade a the written exam taken during the Summer session is valid until the Fall session, and the one take during the Fall one is valid until the Winter one.
ANGELO ESPOSITO Lecturers' profile

Program - Frequency - Exams

Course program
The course is roughly subdivided in four quarters, with about the same duration. 1/4: Finite dimensional Hilbert spaces. Linear operators. Spectral theory. 2/4: Infinite dimensional Hilbert spaces. Spaces of functions. Theory of distributions. Spectral theory. 3/4: Methods for the resolution of ordinary and partial differential equations. Sturm-Louville problem. Green's functions.
Prerequisites
It is mandatory to know linear algebra in finite dimensional vector spaces. In particular, R, R2 and R3. It is mandatory to know total and partial derivatives, as well as integration of functions of one or more variables. It is important to have good understanding of the fundamental theorems on the convergence of series and integrals. It is important to have good understanding on how to perform a change of basis, how to diagonalize and how to solve an eigensystem, at least for the simplest cases of R2 and R3.
Books
Petrini, Pradisi, Zaffaroni, "A Guide to Mathematical Methods for Physicists with Problems and Solutions", World Scientific. Petrini, Pradisi, Zaffaroni, "A Guide to Mathematical Methods for Physicists: Advanced Topics and Applications", World Scientific. Bernardini, Ragnisco, Santini, "Metodi Matematici della Fisica", Carocci Editorie. Zanghì, "Appunti di Metodi Matematici della Fisica", Università di Genova, downloadable from: https://www.ge.infn.it/~zanghi/metodi/ZUL.pdf. Calogero, "Metodi Matematici della Fisica", Sapienza, lecture notes available online.
Frequency
Optional
Exam mode
During the written exam it is allowed to use only one book. Formula sheets and personal notes can be used upon request to the teacher and only under their supervision. To access the oral examination it is necessary to have passed the written one with a grade equal or higher than 18. A passing grade in the written exam allows to take one oral examination only. If the final grade is not passing or it is rejected by the student, the grade of the written part is lost. Only the last submitted written exam is valid, all previous results are deleted. Students can decide not to take the oral examination, in which case, the final grade is the lowest between the result of the written part and 25. Students can retake the exam during the next appeal of the same session. The grade of the written exam remains valid until the Winter session 2024. Hence, the grade a the written exam taken during the Summer session is valid until the Fall session, and the one take during the Fall one is valid until the Winter one.
Lesson mode
In class lectures and recitations. Uniquely blackboard presentations.
  • Lesson code1018973
  • Academic year2025/2026
  • Coursecorso|33588
  • CurriculumFisica
  • Year2nd year
  • Semester2nd semester
  • SSDFIS/02
  • CFU12
  • Subject areaTeorico e dei fondamenti della Fisica