Learning objectives

In this course, the student will explore more advanced topics in mathematical analysis applied to engineering, in particular the theory of Fourier series, integral transforms, functions of a complex variable, and tempered distributions. By the end of the course, the student is expected to have knowledge and understanding of techniques in real and functional mathematical analysis and the ability to use them to solve complex problems, as well as to appropriately use the language and formalism of real and functional mathematical analysis in various engineering and modeling applications.

Prerequisites and basic notions

Basic competencies of Mathematical Analysis, Linear Algebra, and Geometry are required. In particular, the following knowledge is expected:
-- complex numbers; sequences and series of real numbers; limits; differential and integral calculus for functions of one or several variables; first order linear and separable differential equations; second order linear differential equations with constant coefficients;
-- vectors in ℝⁿ, matrices and linear systems; eigenvalues and eigenvectors;
-- euclidean geometry of the plane and three-dimensional space.

Program

1. Complex analysis in one variable: functions of a complex variable, continuity and differentiability in the complex sense, Cauchy-Riemann identity, holomorphic functions. Cauchy integral formula. Complex power series, radius and domain of convergence, absolute and total convergence. Derivative and anti-derivative of a complex power series. Analytic functions, Taylor series. Holomorphic and analytic functions. Zeros, singularities and poles, Laurent series expansion. Decomposition of a holomorphic function. Residues and the residue theorem. Partial fraction decomposition. Integrals with the residue method: integrals of the first, second, third, and fourth kind.
2. Fourier series: trigonometric series, Fourier series, Euler formulas. Elementary harmonics. Fourier series expansion of even and odd functions. Fourier series of the triangular wave, the square wave, and the sawtooth. Pointwise convergence. Fourier series, horizontal and vertical translations, and rescaling. Square-summable functions. L2 theory. Energy of a periodic signal. Convergence in quadratic mean, Bessel inequality, Parseval identity. Half-period extension.
3. Fourier transform: introduction to integral transforms. Fourier transform in L1. Properties. Transform of the Gaussian, of the characteristic function, and of Lorentzian functions. Fourier transform and differential equations. Convolution. Fourier transform of the triangle function. Dirichlet integral and transform of the sinc function. L2 theory.
4. Laplace transform: injectivity of the Laplace transform and inversion formula. Exercises on the Laplace original. Laplace original of rational functions. Application of the Laplace transform to the resolution of linear differential equations.
5. Introduction to the theory of distributions: test functions and the space D of infinitely differentiable functions with compact support. Linear functionals on the space D and linear functionals associated with a locally summable function. The space of distributions D′. Examples. The Dirac delta or evaluation functional. Informal properties of δ. Limits in the sense of distributions. Sequences of functions that converge, in the sense of distributions, to the Dirac δ. Derivative in the sense of distributions: properties. Distributional derivative of the Heaviside function. Properties of the Dirac δ.
6. Elements of zeta transform: discrete signals, definition of unilateral Z transform. Linearity, and transform of significant signals: exponential, binomial, shifts (lag and advance), power, derivation, and cos(ωn).

Bibliography

Didactic methods

All the topics will be illustrated in class. Additional material, such as exercises, lecture notes, and further references, will be available on the Moodle page of the course.

Learning assessment procedures

The exam consists of a written test that requires students to solve some exercises and answer some theoretical questions on definitions, statements, and proofs covered during the course. To pass the examination, a minimum grade of 18/30 must be obtained.
During the exam, the use of any artificial intelligence tool is strictly prohibited. Academic integrity requires students to demonstrate their own level of learning, as only through independent work and critical thinking can they acquire the skills and knowledge that will be assessed.

Evaluation criteria

To pass the exam, the student must demonstrate:
-- to have understood the theoretical notions, showing detailed knowledge of definitions and statements, as well as of some proofs;
-- to be able to apply theory to problem-solving;
-- to be able to express themselves clearly and rigorously, using technical terminology appropriately.

Exam language

Italiano

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