1. The course is designed to provide students enrolled in the Bachelor’s Degree Programme in Electronic Engineering with the advanced mathematical analysis tools that are essential for successfully tackling engineering subjects in the second and third years. The syllabus combines the traditional contents of Mathematical Analysis II (sequences and series of functions, multivariable calculus, differential equations, and vector calculus) with topics traditionally covered in Mathematical Analysis III (the theory of holomorphic functions, Laurent series, the Fourier transform, and the Laplace transform).

   In particular, the learning objectives of the course, formulated according to the Dublin Descriptors, are as follows:

   1. Knowledge and understanding: The student will acquire knowledge and understanding of the tools of advanced mathematical analysis: sequences and series of functions, power and Taylor series, differential and integral calculus in several variables, vector calculus, ordinary differential equations, functions of a complex variable, Laurent series, and Fourier and Laplace transforms. The student will be able to state and understand the main theorems, recognize the underlying mathematical structures, and master the rigorous language of the discipline.

   2. Applying knowledge and understanding: The student will be able to apply the knowledge acquired to the mathematical modelling of engineering problems, including Taylor series expansions of functions for approximation, the solution of differential equations using the Laplace transform, frequency-domain analysis of signals using the Fourier transform, the evaluation of real integrals by means of the residue theorem, and the analysis of LTI systems and electrical circuits.

   3. Making judgements: The student will be encouraged to independently deepen their knowledge, identify the most appropriate mathematical tool for the problem at hand, and critically assess the correctness and completeness of proposed solutions. Constructive interaction among students and continuous dialogue with the instructor will be encouraged.

   4. Communication skills: Through attending lectures, participating in problem-solving sessions, and reading the recommended textbooks, the student will learn to communicate acquired mathematical results rigorously and clearly, both in written and oral form, while mastering the formal language of the discipline.

   5. Learning skills: The student will be guided in consolidating their study methods and developing the ability to approach new topics independently, identify the necessary prerequisites, and transfer the skills acquired to subsequent courses (Electrical Engineering, Signal Theory, Automatic Control, and Electrical Communications).

The course is delivered through theory lectures and exercise sessions, using the blackboard. Multimedia aids may occasionally be used. Should the course be delivered in blended or remote mode, the necessary changes may be introduced with respect to what has been stated above, in order to comply with the program set out in the Syllabus.

Student office hours may also be held remotely, by appointment.

It is essential to master all the concepts and types of exercises in Calculus I. In particular, the student must be able to:

▸     calculate limits of functions and numerical sequences

▸     recognize points of continuity of functions and classify their singularities

▸     calculate derivatives of functions of a real variable and identify their stationary points

▸     study the behavior of a numerical series (ratio test, root test, Leibniz test, comparison test)

▸     calculate definite and indefinite integrals, including improper integrals

▸     determine the general integral of a first-order differential equation

Basic knowledge of analytic geometry in the plane and in space, as well as the fundamentals of vector space theory (Linear Algebra and Geometry), is also useful.

Module 1 - Sequences and Series of Functions, Power Series and Taylor Series (8 lecture hours + 5 exercise hours)

• Sequences of real-valued functions of a real variable: pointwise and uniform convergence
• Characterization of uniform convergence
• Cauchy convergence criterion for sequences (pointwise and uniform)
• Theorems on continuity, differentiability, and passage to the limit under the integral sign
• Series of real-valued functions: pointwise, uniform, absolute, and total convergence
• Cauchy criterion and Weierstrass theorem for series of functions
• Comparison of the different types of convergence. Theorems on continuity, differentiability, and integration of series
• Power series: radius of convergence, Cauchy–Hadamard theorem, Abel's theorem
• Properties of the sum function of a power series (continuity, differentiability, integrability)
• Taylor series: conditions for Taylor-expandability; standard expansions (exponential, sine, cosine, logarithm, binomial)
• Fourier series: Fourier coefficients, pointwise convergence (Dirichlet conditions)

Module 2 — Differential and Integral Calculus in Rⁿ (15 lecture hours + 10 exercise hours)

• Euclidean spaces; functions between Euclidean spaces; topology in Rⁿ
• Limits and continuity of functions of several variables; Weierstrass theorem and existence-of-zeros theorem
• Partial derivatives and directional derivatives; gradient, Jacobian matrix, Hessian matrix
• Differentiability; total differential theorem
• Chain rule; first- and second-order Taylor formulas
• Zero-gradient theorem; positively homogeneous functions (Euler's identity)
• Implicit Function Theorem of U. Dini (scalar implicit function and vector case)
• Relative maxima and minima: necessary condition (Fermat); sufficient second-order conditions
• Finding absolute extrema
• Constrained optimization: method of Lagrange multipliers
• Lebesgue integral in Rⁿ: measure, measurable functions, Fubini's and Tonelli's theorems
• Reduction formulas for double and triple integrals; change of variables
• Polar, cylindrical, and spherical coordinates

Module 3 — Vector Calculus and Field Theory (8 lecture hours + 5 exercise hours)

• Curves in Rⁿ: parametrization, tangent and normal vectors, rectifiability, arc length
• Scalar and vector line integrals
• Linear differential forms: exactness, closedness, circulation; Poincaré theorem
• Conservative vector fields and scalar potentials
• Differential operators: gradient, divergence, curl, Laplacian
• Regular domains; Gauss–Green formulas in the plane
• Surfaces in R³: parametrization, normal vector, surface integrals
• Gauss divergence theorem in space
• Stokes' theorem

Module 4 — Ordinary Differential Equations (10 lecture hours + 5 exercise hours)

• ODEs of order n; Cauchy problem: local and global existence and uniqueness
• Separable, homogeneous, first-order linear, and Bernoulli equations
• Linear systems of ODEs: structure of the solution set, Wronskian matrix
• Linear systems with constant coefficients: eigenvalue and eigenvector method
• Linear ODEs with constant coefficients: homogeneous and particular solutions
• Method of variation of parameters (Lagrange); Euler equation

Module 5 — Complex Variable Functions and Laurent Series (10 lecture hours + 5 exercise hours)

• The field of complex numbers: topology of the complex plane
• Functions of a complex variable: limit, continuity, complex differentiability
• Cauchy–Riemann equations; harmonic conjugate functions
• Holomorphic (analytic) functions: definition and fundamental properties
• Elementary functions in the complex field: exponential, trigonometric and hyperbolic functions, complex logarithm, powers
• Complex line integrals; Darboux theorem
• Cauchy–Goursat theorem; primitive of a holomorphic function; Morera's theorem
• Cauchy's integral formulas and consequences (higher-order derivatives)
• Analyticity of holomorphic functions; Hermite–Liouville theorem
• Fundamental theorem of algebra
• Taylor series in the complex field; power series and their convergence
• Laurent series: Laurent's theorem, regions of convergence (annular regions)
• Isolated singularities: removable singularities, poles of order n, essential singularities
• Zeros of a holomorphic function; identity theorem for holomorphic functions
• Residue; residue theorem
• Applications: calculation of improper real integrals using the residue theorem

Module 6 — Fourier Transform (5 lecture hours + 2 exercise hours)

• Introduction to functions of bounded variation and absolutely continuous functions
• Fourier transform of integrable functions: definition and domain of existence
• Properties: linearity, time and frequency translation, scaling, conjugation, symmetry (even/odd functions)
• Riemann–Lebesgue theorem; continuity of the Fourier transform
• Fourier transform of derivatives and integrals
• Convolution of integrable functions; Fourier transform of convolution; multiplication formula
• Fourier inversion; adjoint transform
• Parseval's theorem (Parseval's equality)

Module 7 — Laplace Transform (5 lecture hours + 2 exercise hours)

• Laplace transform of locally integrable functions: definition, abscissa of convergence and absolute convergence
• Holomorphicity of the transform in the half-plane of absolute convergence; uniform convergence
• Operational properties: linearity, time translation, translation in the complex plane, scaling
• First and second fundamental formulas (transform of the nth derivative and of the integral)
• Convolution of locally integrable functions; Laplace transform of convolution
• Inversion of the transform: inverse transformation of rational functions using partial fractions
• Applications to solving ODEs with initial conditions and systems of ODEs

The exam aims to verify the achievement of the expected learning outcomes, with particular attention to knowledge of the fundamental theoretical concepts and the ability to apply them to solve problems typical of electronic engineering.

The exam consists of a written test lasting two hours, followed by an oral examination.

The written test is divided into two parts:

–       Part A (theory): 2 definitions and 2 theoretical questions. Minimum requirement: correctly answer at least 1 definition and 1 theoretical question.

–       Part B (exercises): 4 exercises (one for each macro-topic: series of functions/calculus in Rⁿ, vector calculus/ODEs, complex variable, transforms). Minimum requirement: correctly complete at least 2 exercises.

The oral examination covers the entire course syllabus. It may be taken after passing the written test and allows the grade to be improved. An insufficient performance may result in a reduction of the written test grade or, in more serious cases, the annulment of the exam.

The evaluation criteria for the oral exam include:

• the relevance of the answers to the questions asked;

• the quality and accuracy of the content presented;

• the ability to connect the various topics of the program;

• the ability to provide relevant examples and applications;

• the student's command of technical and mathematical language;

• the student's overall expressive ability.

Theory: radius of convergence of a power series; conditions for expandability into a Taylor series; Cauchy-Riemann conditions; residue theorem; classification of isolated singularities; properties of the Fourier transform; initial value and final value theorem for the Laplace transform; Fubini's theorem; structure of the solutions of a system of linear ODEs.

Exercises: determining the radius of convergence and the sum of a power series; Taylor/Laurent series expansion of a given function; computing improper real integrals using the residue theorem; solving ODEs with initial conditions using the Laplace transform; frequency analysis of a signal using the Fourier transform; unconstrained and constrained optimization of functions of several variables; computing multiple integrals with change of variables.