#
Mathematical methods in physics

A.Y. 2021/2022

Learning objectives

Aim of the course is to introduce the students to the methods of complex analysis and functional analysis. In spite of its introductory level, the course also tries to be rigorous, and many proofs are included when significant. Very important points of the program are:

-Holomorfic functions with examples of maps, Taylor series, the Cauchy theorem and its use, isolated singurities and Laurent's expansion, the residue theorem and integration in the complex plane. Analytic continuation.

- Banach and Hilbert spaces, examples of spaces of functions. Introduction to the theory of linear operators on Hilbert spaces.

-Fourier series and Fourier and Laplace transforms.

-Introduction to the theory of tempered distributions.

-Holomorfic functions with examples of maps, Taylor series, the Cauchy theorem and its use, isolated singurities and Laurent's expansion, the residue theorem and integration in the complex plane. Analytic continuation.

- Banach and Hilbert spaces, examples of spaces of functions. Introduction to the theory of linear operators on Hilbert spaces.

-Fourier series and Fourier and Laplace transforms.

-Introduction to the theory of tempered distributions.

Expected learning outcomes

At the end of the course the student will be able to:

1) use and manipulate complex numbers along with their geometric meaning and representation, carry out arithmetic and algebraic operations in the complex plane, study geometric mappings.

2) will be able to carry out studies of functions (single and multi-valued) in the complex plane

3) will be able to compute integrals in the complex plane, with integration techniques based upon Cauchy's theorem and the calculus of residues

4) will be able to understand and utilize basic concepts about Hilbert and Banach spaces, and orthonormal functions (Hermite, Legendre)

5) will be able to understand the main properties of linear bounded operators such as projections, isometries, unitary operators, functions of an operator, self-adjointness (with the extension to unbounded operators). Will be able to apply the theory to finite matrix operators, and in some infinite-dimensional cases.

6) will have knowledge of Fourier series, their point and norm convergence, and evaluate the series for simple functions.

7) will have knowledge of the Fourier (and Laplace) transform in L1 and L2, and of Riemann-Lebesgue's theorem. He will evaluate the main Fourier transforms, also by techniques of integration in the complex plane.

7) will have knowledge of the basic theory of tempered distributions, the most important ones (delta, theta, principal part), their

derivative and Fourier transform, and applications (Sokhotskii-Plemelj identity).

1) use and manipulate complex numbers along with their geometric meaning and representation, carry out arithmetic and algebraic operations in the complex plane, study geometric mappings.

2) will be able to carry out studies of functions (single and multi-valued) in the complex plane

3) will be able to compute integrals in the complex plane, with integration techniques based upon Cauchy's theorem and the calculus of residues

4) will be able to understand and utilize basic concepts about Hilbert and Banach spaces, and orthonormal functions (Hermite, Legendre)

5) will be able to understand the main properties of linear bounded operators such as projections, isometries, unitary operators, functions of an operator, self-adjointness (with the extension to unbounded operators). Will be able to apply the theory to finite matrix operators, and in some infinite-dimensional cases.

6) will have knowledge of Fourier series, their point and norm convergence, and evaluate the series for simple functions.

7) will have knowledge of the Fourier (and Laplace) transform in L1 and L2, and of Riemann-Lebesgue's theorem. He will evaluate the main Fourier transforms, also by techniques of integration in the complex plane.

7) will have knowledge of the basic theory of tempered distributions, the most important ones (delta, theta, principal part), their

derivative and Fourier transform, and applications (Sokhotskii-Plemelj identity).

**Lesson period:**
Second semester

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### CORSO A

Responsible

Lesson period

Second semester

In case of emergency, the course is delivered online via zoom, in real time. The program is essentially the same, with drills and online tutoring. Recorded lesson and notes are timely made available in ARIEL.

The exam (online with video on) consists of three problems to be solved on paper in 30 minutes each, and mailed in photo to the teacher before di next problem. The maximum score is 25. The student who is sufficient in the test may either accept the grade, or ask for a supplementary oral exam (via zoom) of about 30 minutes.

The exam (online with video on) consists of three problems to be solved on paper in 30 minutes each, and mailed in photo to the teacher before di next problem. The maximum score is 25. The student who is sufficient in the test may either accept the grade, or ask for a supplementary oral exam (via zoom) of about 30 minutes.

**Course syllabus**

Complex analysis: holomorphic functions, conformal maps, problems in 2D electrostatics, complex integral, Cauchy transform, index function, Cauchy theorems, power and Laurent series, isolated singularities, residue theorem, analytic continuation, Gamma function.

Functional analysis: Hilbert spaces with examples, orthogonal polynomials, orthonormal bases (Hermite, Legendre), basics of theory of linear bounded oparators (adjoint, projection, unitary, function of operator), adjoint of unbounded operators with examples, Fourier series (point and norm convergence), space S of rapidly decreasing functions and S' of tempered distributions, Fourier transform in S, S', L1 and L2, inversion and convolution. Riemann-Lebesgue theorem.

Functional analysis: Hilbert spaces with examples, orthogonal polynomials, orthonormal bases (Hermite, Legendre), basics of theory of linear bounded oparators (adjoint, projection, unitary, function of operator), adjoint of unbounded operators with examples, Fourier series (point and norm convergence), space S of rapidly decreasing functions and S' of tempered distributions, Fourier transform in S, S', L1 and L2, inversion and convolution. Riemann-Lebesgue theorem.

**Prerequisites for admission**

Linear algebra (real and complex vector spaces; Hermitian, unitary and orthogonal matrices; eigenvalues and eigenvectors; Cayley-Hamilton theorem). Lebesgue integral. Sequences and series, sequences of functions, point and uniform convergence. Metric and normed spaces. Ordinary differential equations.

**Teaching methods**

Lesson and drills in classroom. Tutoring is normally available.

**Teaching Resources**

Textbook Mathematical Methods for Physics (L. Molinari) (printed by CUSL) and a collection of exam tests and exercises (some with solution) are available in ARIEL, with links to textbooks of the digital library of UniMi.

Useful textbooks: Bak and Newman, Complex Analysis, Springer (available online in digital library of Milan's University)

Kolmogorov and Fomine, Elements of the theory of functions and functional analysis, reprint Dover.

Useful textbooks: Bak and Newman, Complex Analysis, Springer (available online in digital library of Milan's University)

Kolmogorov and Fomine, Elements of the theory of functions and functional analysis, reprint Dover.

**Assessment methods and Criteria**

The written exam of duration 3H, consists of 3/4 exercises. An exercise is positively evaluated if the steps are also adequately commented. During the test, the student may consult the textbooks made available.

The grades are published anonymous, with reference to the matriculate number. The student who tries a new test, after one whose evaluation he has not yet been accepted or refused, automatically refuses the grade of the previous test.

Il the written exam is evaluated not less than 25/30, the student may decide to have an oral exam, that begins with a presentation of a topic chosen by the student, followed by questions to ascertain the knowledge of fundamental topics of the program.

The grades are published anonymous, with reference to the matriculate number. The student who tries a new test, after one whose evaluation he has not yet been accepted or refused, automatically refuses the grade of the previous test.

Il the written exam is evaluated not less than 25/30, the student may decide to have an oral exam, that begins with a presentation of a topic chosen by the student, followed by questions to ascertain the knowledge of fundamental topics of the program.

FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 7

Practicals: 24 hours

Lessons: 40 hours

Lessons: 40 hours

Professors:
Fratesi Guido, Molinari Luca Guido

### CORSO B

Responsible

Lesson period

Second semester

Lectures notes prepared by the lecturer, which include all steps in the mathematical derivations, will be made available to the student for each lecture. Videos of the lectures will be made available online for the students who cannot attend the lectures in presence.

**Course syllabus**

Complex analysis: holomorphic functions, conformal maps, problems in 2D electrostatics, Riemann sheets analysis of multi-valued functions, complex integral, Cauchy transform, index function, Cauchy theorems, power and Laurent series, isolated singularities, residue theorem, analytic continuation, Gamma function.

Functional analysis: Hilbert spaces with examples, orthogonal polynomials, orthonormal bases (Hermite, Legendre), basics of theory of linear bounded oparators (adjoint, projection, unitary, function of operator), adjoint of unbounded operators with examples, Fourier series (point and norm convergence),

space S of rapidly decreasing functions and S' of tempered distributions, Fourier transform in S, S', L1 and L2, inversion and convolution. Riemann-Lebesgue theorem. Theory of distributions and Plemelj identity.

Functional analysis: Hilbert spaces with examples, orthogonal polynomials, orthonormal bases (Hermite, Legendre), basics of theory of linear bounded oparators (adjoint, projection, unitary, function of operator), adjoint of unbounded operators with examples, Fourier series (point and norm convergence),

space S of rapidly decreasing functions and S' of tempered distributions, Fourier transform in S, S', L1 and L2, inversion and convolution. Riemann-Lebesgue theorem. Theory of distributions and Plemelj identity.

**Prerequisites for admission**

Linear algebra (real and complex vector spaces; Hermitian, unitary and orthogonal matrices; eigenvalues and eigenvectors; Cayley-Hamilton theorem). Lebesgue integral.

Sequences and series, sequences of functions, point and uniform convergence. Metric and normed spaces. Ordinary differential equations.

Sequences and series, sequences of functions, point and uniform convergence. Metric and normed spaces. Ordinary differential equations.

**Teaching methods**

Lessons and supervisions in classroom with all derivations at the blackboard and/or AV systems. Tutoring is normally available.

**Teaching Resources**

Reference textbooks:

- C. W. Wong "Introduction to Mathematical Physics", Oxford University Press

- K. Cahill, "Physical Mathematics", Cambridge University Press

- G. B. Arfken, H.J. Weber, "Mathematical methods for physicists", Elsevier

- G. Cicogna, "Metodi matematici della fisica", Springer

- C. W. Wong "Introduction to Mathematical Physics", Oxford University Press

- K. Cahill, "Physical Mathematics", Cambridge University Press

- G. B. Arfken, H.J. Weber, "Mathematical methods for physicists", Elsevier

- G. Cicogna, "Metodi matematici della fisica", Springer

**Assessment methods and Criteria**

Written exam of duration 3H, with 4-5 exercises. An exercise is positively evaluated if the steps are also adequately commented. Normally, one correct exercise guarantees sufficiency. During the test, the student may consult the textbooks made available. The grades are published anonymously, with reference to the matriculate number. The student who tries a new test, after one whose evaluation he has not yet accepted or refused, automatically refuses the grade of the previous test.

Il the written exam is evaluated greater than 24/30, the student may decide to have an oral exam, that begins with presentation of a topic chose by the student, followed by questions to ascertain the knowledge of fundamental topics of the program.

Il the written exam is evaluated greater than 24/30, the student may decide to have an oral exam, that begins with presentation of a topic chose by the student, followed by questions to ascertain the knowledge of fundamental topics of the program.

FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 7

Practicals: 24 hours

Lessons: 40 hours

Lessons: 40 hours

Professor:
Zaccone Alessio

### Corso C - Edizione erogata in lingua inglese

Responsible

Lesson period

Second semester

FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 7

Practicals: 24 hours

Lessons: 40 hours

Lessons: 40 hours

Professor:
Röntsch Raoul Horst

Educational website(s)

Professor(s)