# 30591 - ELEMENTS OF REAL AND COMPLEX ANALYSIS

Course taught in English

Go to class group/s: 31

ELIA BRUE'

A sound knowledge of the main tools of calculus (limits, series, derivatives, integrals, also in the multivariable case) and of linear algebra, basic concepts of topology.

The course will provide the basic foundations of Complex and Fourier Analysis, introducing some fundamental tools for signal theory. Complex analysis deals with complex functions of a complex variable and enlights remarkable links between complex differentiability, power series in the complex plane, line integral representations, conformal maps, and harmonic functions. Its main results are particularly important to understand power series (which lie at the core of functional calculus for linear operators), to capture the geometric properties of conformal transformations in the plane, to obtain integral representations of harmonic functions, and to understand the Laplace and the discrete Zeta and Fourier transforms. Fourier analysis is one of the most powerful tools to analyze functions and it is the basic building block of signal theory. Starting from Fourier series dealing with periodic signals, which can be interpreted as an orthonormal decomposition in Hilbert spaces, its range of applications is considerably expanded by the Fourier and the Laplace transforms, which cover the case of general signals. The course is meant to round up an adequate undergraduate preparation in Mathematical Analysis and to give students a hint on more advanced issues, that find surprising and remarkable applications in several theoretical and applied fields.

The course is divided into two parts.

After a quick review of complex numbers, the first part of the course will concern with:

- holomorphic functions,

- power series expansions in the complex field,

- line integrals,

- index/winding number of a curve about a point,

- singularities and residuals, Laurent expansions

- the Theorems of Riemann and Cauchy, the residue Theorem,

- applications*: the argument and the maximum principles, the fundamental theorem of Algebra, the Zeta transform.

After a brief and informal recap of some properties of Hilbert spaces and the Lebesgue space L2, the second part of the course will be devoted to

- Fourier series in the framework of orthogonal systems of signals,

- properties and convergence of Fourier series,

- Fourier transform,

- applications*: Laplace transform, solutions to differential equations, Poisson summation formula and sampling theorem, the Heisenberg uncertainty principle.

(The choice of starred* topics will be adjusted according to the time available)

Understand the basic facts, tools, and techniques of complex analysis: recognize holomorphic functions and their singularities, identify power series and their convergence, describe their main properties, recognize closed circuits and their winding numbers in the complex plane, explain the scope of the basic integral theorems and representation formulae.

Understand the basic structure of Hilbert spaces, the use of scalar products, and of orthonormal systems and expansions.

Know the basic properties of periodic signals and the trigonometric basis, understand the meaning of Fourier expansion, reproduce basic series expansions, and describe their convergence properties.

Understand Fourier transform, its inversion, and its link with Fourier series and Laplace transform.

Manipulate simple power series in the complex plane.

Analyze singularities, compute residuals, and evaluate simple curvilinear integrals.

Use the power series expansions of fundamental functions. Use rational functions.

Solve simple exercises on holomorphic functions.

Compute Fourier series expansions of simple signals. Use Fourier series expansions for solving differential equations with periodic solutions. Estimate the behavior of the Fourier coefficients.

Compute the Fourier/Laplace transform of simple signals. Interpret the Fourier transform and its behavior. Use simple relations between a function and its Fourier/Laplace transform.

- Face-to-face lectures
- Online lectures
- Exercises (exercises, database, software etc.)

Online lectures have the same conceptual role as face-to-face lectures. The actual blend of face-to-face lectures and online lectures will mainly depend on external constraints.

Exercise sessions (again: both face-to face and online) are dedicated to the application of the main theoretical results obtained during lectures to problems and exercises of various nature.

Continuous assessment | Partial exams | General exam | |
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x | x |

Students will be evaluated on the basis of written exams, which can be taken in one of the two following ways.

The exam can be split in two partial exams. Each partial may contain multiple-choice questions and/or open-answer questions; each partial weighs for one-half of the final mark. Multiple-choice questions mainly aim at evaluating the knowledge of the fundamental mathematical notions and the ability to apply these notions to the solution of simple problems and exercises, while open-answer questions mainly aim at evaluating:

- The ability to articulate the knowledge of mathematical notions in a conceptually and formally correct way, adequately using definitions, theorems and proofs.
- The ability to actively search for deductive ideas that are fit to prove possible links between the properties of mathematical objects.
- The ability to apply mathematical notions to the solution of more complex problems and exercises.

The exam can also be taken as a single general exam, which may contain multiple-choice questions and open-answer questions. The general exam covers the whole syllabus of the course and it can be taken in one of the four general sessions scheduled in the academic year. This option is mainly meant for students who have withdrawn from the two-partials procedure or could not follow it. Each type of question contributes in a specific way to the assessment of the students' acquired knowledge. Multiple-choice questions mainly aim at evaluating the knowledge of the fundamental mathematical notions and the ability to apply these notions to the solution of simple problems and exercises, while open-answer questions mainly aim at evaluating:

- The ability to articulate the knowledge of mathematical notions in a conceptually and formally correct way, adequately using definitions, theorems and proofs.
- The ability to actively search for deductive ideas that are fit to prove possible links between the properties of mathematical objects.
- The ability to apply mathematical notions to the solution of more complex problems and exercises.

We will take care to obtain final grades whose distribution follows the grade distribution that is recommended by Università Bocconi.

Steven Krantz: *A guide to complex variables*

The Mathematical Association of America, 2008

Henri Cartan: *Elementary theory of analytic functions of one or several complex variables*

Dover 1995

Elias Stein, Rami Shakarchi: *Fourier Analysis, an introduction*

Princeton University Press, 2002

Pierre Bremaud: *Mathematical Principle of Signal Processing*

Springer, 2002