EDUCATIONAL GOALS
The course presents the basic elements of Mathematical Analysis 2. The educational aims of the course include the acquisition of results and demonstrative techniques, as well as the ability to use the related calculation tools.
Knowledge and understanding of the following aspects is expected:
1) Indefinite and definite integrals.
2) Numerical series.
3) Functions of two variables.
4) Differential equations.
5) Series of functions.
6) Complex analysis.
Knowledge and understanding of applicative type are expected on the following aspects:
1) Ability to perform calculations with integrals and series.
2) Formalization of procedures to solve optimization problems in the case of two-variable functions.
3) Know how to solve a differential equation.
4) Ability to understand the convergence sets of particular series of functions.
5) Knowing how to solve complex analysis problems.

PROVIDING EDUCATION
N. 48 ON-LINE VIDEO LESSONS (N. 12 EDUCATIONAL UNITS - DURATION OF TWO HOURS FOR EACH CFU).

INTERACTIVE EDUCATION
N. 2 INTERACTIVE LESSONS PER CFU.
N. 5 THEMATIC DISCUSSIONS ON THE EDUCATIONAL FORUM (TOPIC) AND N. 2 POSTS PER CFU AS PER THE PQA EDUCATIONAL GUIDELINES.
N. 2 E-TIVITIES EVERY 5 CFU.
N. 2 TESTS FOR EACH CFU WITH 8 MULTIPLE CHOICE QUESTIONS.

COURSE PROGRAM
Module 1. Indefinite and definite integrals:
Primitive and indefinite integral. Immediate integrals of elementary and compound functions. Integration rules and methods. Integrals by parts and by substitution. Definite integral and its geometric meaning. Mean theorem. Integral function and fundamental theorem of integral calculus. Fundamental formula of integral calculus.

Module 2. Numerical series:
Connection between numerical sequences and numerical series. Convergent, divergent and indeterminate series. Geometric, harmonic and telescopic series. Convergence criteria for series with positive terms: comparison, ratio and root criterions. Leibniz criterion for the convergence of alternating series.

Module 3. Functions of two variables:
Domains and topology for functions of two variables. Limits, continuity and derivability. Differentiability. Relative maxima and minima for functions of two variables: generality; first order necessary condition. Second order sufficient condition.

Module 4. Differential equations:
First and second order Cauchy problem. Cauchy's theorem for first order differential equations. First order differential equations: linear ones with variable coefficients; with separable variables; homogeneous; of type f (ax + by). Second order differential equations with constant coefficients.

Module 5. Series of functions:
Pointwise, absolute, uniform and total convergence of a series of functions. Power series: generalities; convergence radius and convergence set; Cauchy - Hadamard's theorem; D'Alembert's theorem. Fourier series: generalities and conditions for the developability; computation of Fourier coefficients for functions of any type, even functions and odd functions. Bessel inequality and Riemann - Lebesgue lemma. Pointwise convergence theorem. Uniform convergence theorem. Parseval equality.

Module 6. Complex analysis:
Elementary functions of complex variable. Limits, continuity and derivability. Holomorphy. Points of singularity and their classification. Integration in the complex field. Cauchy's theorem and integral formula. Laurent series and classification of singularity points. Residues and residue theorem.

METHOD OF VERIFYING PROFIT IN ITINERE
The degree of student learning is constantly monitored through verification tools and methodologies.
In particular, in order to make the verification and certification of the training results feasible, the teacher and the tutor will consider the following aspects:
1) automatic tracking of training activities by the system - reporting;
2) didactic and technical monitoring (in terms of quantity and quality of interactions, compliance with didactic deadlines, delivery of the planned papers, etc.);
3) ongoing training checks, also for self-assessment (eg multiple choice tests, true/false, sequence of questions with varying difficulty, simulations, mind maps, works, group projects, etc.);
4) final exam, during which it is considered and valued not only the performance during the exam, but also the work done online (remote activities, quantity and quality of online interactions, etc.) .
In this framework, the evaluation takes into account several aspects:
a) the result of a certain number of intermediate tests, if foreseen (in terms of online tests, development of works, etc.);
b) the quality and quantity of participation in online activities (frequency and quality of interventions that can be monitored through the platform);
c) the results of the final exam.
It is specified that the final test is only of the "written" type. The oral test is optional and is useful only and exclusively to refine the evaluation of the written test. The topics of the oral exam, if carried out, are agreed between the teacher and the student.
All the collected data will be, in general, evaluated by the teacher to give the student an evaluation that is objective and consistent with the university's objectives, taking into account both the summative aspects and the formative aspects.

ASSESSMENT METHOD AND AIMS OF THE FINAL EXAM
The access to the exam is subordined to the recognition of attendance, which will be recognized, at the booking time for the exam, by an appropriate certificate, that will certify the development of the didactic verification activities in progress and at the level of the work made in the various exercises.
The exam will consist of a written test, with an optional oral one. The final evaluation will be expressed out of 30, with possible honors depending on the degree of maturity reached by the student.
The exam aims at the didactic aims described as follows. In particular, the student will have to:
a) be able to recognize a possible methodology to solve an indefinite or definite integral;
b) know how to study the features of a numerical series;
c) be able to identify possible relative maxima/minima for functions of two variables;
d) know how to solve various types of differential equations;
e) be able to study the convergence of suitable series of functions;
f) know how to solve problems of Complex Analysis.
The written test foresees a score for all the previous aims, from point a) to point f). Each goal has a variable score. Honors will be awarded in those particular cases in which particular maturity is demonstrated in solving the exam questions.

KNOWLEDGE AND UNDERSTANDING IN TERMS OF EXPECTED OUTCOMES (DUBLIN DESCRIPTOR NO. 1)
The knowledge of the theory of integrals will allow to properly estimate the volume of traffic flows in telecommunications networks.
The knowledge of numerical series theory will allow to define convergence techniques for ad hoc numerical schemes for the simulation of data traffic networks.
The knowledge of the theory of functions of two variables will allow the student to define appropriate optimization techniques for traffic performance in telecommunications networks.
Knowledge of the theory of differential equations will allow a simpler understanding of variational phenomena within the contexts of Information and Communication Technology systems.
The knowledge of the theory on the series of functions will allow to develop suitable methodologies for the management of the convergence of numerical traffic simulation schemes.
The knowledge of the theory of complex analysis will allow to manage models and methods for the analysis of signals.

SKILLS TO APPLY KNOWLEDGE AND UNDERSTANDING IN TERMS OF EXPECTED OUTCOMES (DUBLIN DESCRIPTOR NO. 2)
The student will be able to use the knowledge related to the theory of integration, numerical series and series of functions to solve analytical/numerical problems commonly encountered in normal traffic networks.
The student will be able to use knowledge on functions of several variables to formulate traffic optimization problems in traffic networks.
The student will be able to use the knowledge of differential equations to understand the speed of the dynamics of data variation in telecommunications networks.
The student will be able to use the knowledge of Complex Analysis to define the general features for the design of data transmission systems.

Paolo Marcellini, Carlo Sbordone, Elementi di Analisi Matematica 2, Versione semplificata per i nuovi corsi di laurea, Liguori Editore, 2016.
Ciro D’Apice, Tiziana Durante, Rosanna Manzo, Verso l’esame di Matematica 2, Maggioli Editore, 2015.
Additional teaching material will be available in the section dedicated to teaching within the university platform.