# MATHEMATICAL METHODS IN PHYSICS

*Last update 14/06/2021 15:08*

OVERVIEW

This course aims to provide adequate knowledge of the most advanced mathematical tools necessary to tackle the study of the physics courses of the third year of the three-year degree. Central themes of the course are the techniques of complex analysis and Fourier and Laplace transforms which will be useful both for subsequent theoretical courses and for courses in experimental physics.

## AIMS AND CONTENT

LEARNING OUTCOMES

The aim of the course is to provide students with the advanced mathematical tools used in modern physics: complex variable functions, Fourier and Laplace transforms, Hilbert spaces and classical partial differential equations of mathematical physics.

AIMS AND LEARNING OUTCOMES

The course has as its main objective the acquisition of basic knowledge and skills related to advanced mathematical tools that have general application in Physics. Particular attention is given to the understanding of the arguments, to the rigor in the presentation of concepts and reasonings, to the applicative aspects of the theoretical tools developed.

We want to provide a solid preparation with a broad spectrum of basic knowledge relating to mathematical tools relevant to physical applications, with the aim of achieving a good ability to apply knowledge and understanding; in particular, to be able to produce simple rigorous proofs of mathematical results not identical to those already known, but clearly related to them, to be able to mathematically formalize problems of moderate difficulty, in order to facilitate their analysis and resolution, to be able to independently read and understand basic texts of Applied Mathematics to physical problems. The presentation of the contents will be carried out in such a way as to improve the student's ability to recognize rigorous demonstrations and identify fallacious reasoning. Particular attention will be paid to the ability to identify the physical motivations that suggest appropriate mathematical structures. In addition, the presentation of the topics will be carried out in such a way as to allow the acquisition of a good ability to communicate problems, ideas and solutions regarding Mathematical Methods for Physics, both in written and oral form. Topics to be explored, closely related to teaching, will be indicated in order to stimulate the student's ability to learn independently.

PREREQUISITES

Contents of Analysis, algebra and geometry carried out in the first two years.

Teaching methods

Both the lessons and the exercises are carried out on the blackboard. Students are always invited to actively participate by asking questions, proposing solutions to the proposed problems. The active involvement of students probably contributes to reducing the time and difficulties related to the study of the topics presented in the course.

SYLLABUS/CONTENT

**Theory of analytic functions**. Algebra and geometry of complex numbers. Analytical and geometric characterization of analytic functions. Cauchy-Riemann conditions. Sequences and series of complex numbers and of functions of complex variables. Exchange of limits and passage to the limit under the derivative sign. Power series in the complex field: radius of convergence. Elementary functions in the complex plane. Integrate them in the complex field. Cauchy's theorem, Cauchy's formula and its applications. Taylor series and its radius of convergence. Isolated singular zeros and points of functions of complex variable. Liouville's theorem. Fundamental theorem of algebra. Morera's theorem. Laurent series. Isolated and essential zeros and singularities. Residues and residue theorem. Classification and general properties of analytic functions. Jordan's lemma and its applications. Singularities on the path of integration. Multi-functions and their properties. Analytical continuation. Analytical continuation of the Fourier transform and its relationship with the Laplace transform. Complex inversion formula of the Laplace transform.

**Fourier analysis and applications**. Generalities on distributions: space of test functions. Definition and properties of the Dirac delta. Integral and derivative of the Dirac delta. The Fourier transform in L_1 and in L_2. General properties of the Fourier transform. properties of the convolution product. The Fourier-Plancherel transform as a unitary operator on L_2 (R). Fourier transform of the Dirac delta. Applications of the Fourier transform to the solution of differential equations. Heat kernel. Laplace transform: definition, properties and examples. Applications.

Vector spaces in infinite dimension. Complete orthonormal systems in infinite dimensional Euclidean spaces. Fourier coefficients. Bessel inequality and Parseval equality. The Hilbert space l_2. Banach spaces: Complete orthonormal systems in L_2 [a, b]. Fourier series: uniform and pointwise convergence.

RECOMMENDED READING/BIBLIOGRAPHY

1] Notes provided by the teacher

2] Neeham, Visual complex analysis

3] Körner, Fourier Analysis

## TEACHERS AND EXAM BOARD

## LESSONS

Teaching methods

Both the lessons and the exercises are carried out on the blackboard. Students are always invited to actively participate by asking questions, proposing solutions to the proposed problems. The active involvement of students probably contributes to reducing the time and difficulties related to the study of the topics presented in the course.

ORARI

L'orario di tutti gli insegnamenti è consultabile su EasyAcademy.

## EXAMS

Exam description

The exam consists of a written test and an interview. The written test consists of some problems that cover a large part of the course contents. The student is then given the freedom to choose between two types of oral: a shorter oral, aimed at consolidating the grade obtained in the written test, and a longer oral in which the change in grade can be significant. This modality is used to allow the student, aware of having achieved a good preparation, to be able to recover any shortcomings of the written test.

Assessment methods

The written test is aimed at verifying the ability to solve specific problems similar to those discussed in the course, but original. The difficulty of the test is graduated, so that it is possible to separate the assessment of basic elementary knowledge, sufficient to pass the test, from the assessment of more advanced skills. Both in the short and in the long form of the oral one always starts from the written test, in order to ascertain the types of errors, the real mastery by the student of the skills required on the theory topics of the written test and highlight any lack of preparation. The long oral exam continues with the assessment of skills on other topics covered in the course. In both cases, the exam is aimed at ascertaining the degree of achievement of the training objectives, in a graded form. In both cases, particular attention is paid to the student's ability to recognize rigorous reasoning, to identify fallacious reasoning and to understand the physical motivations that suggest appropriate mathematical structures.