Physics

LORENZO CAPRINI
ANGELO ESPOSITO

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

The aim of the course is to introduce the mathematical tools underlying the concepts of modern physics. By the end of the course, students will have understood and applied various concepts related to finite- and infinite-dimensional Hilbert spaces, discrete and continuous Fourier analysis, and methods for solving ordinary and partial differential equations.

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.

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.

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.

In class lectures and recitations. Uniquely blackboard presentations.

Optional

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.

Any problem or question concerning the topics taught during the lectures.
Examples of problems and past exams will be provided during the course.