Mathematics III

Ingegneria Elettronica Mathematics III

0612400003
DIPARTIMENTO DI INGEGNERIA INDUSTRIALE
EQF6
ELECTRONIC ENGINEERING
2022/2023

OBBLIGATORIO
YEAR OF COURSE 2
YEAR OF DIDACTIC SYSTEM 2018
AUTUMN SEMESTER
CFUHOURSACTIVITY
660LESSONS
ExamDate
MATEMATICA III17/02/2023 - 09:00
MATEMATICA III17/02/2023 - 09:00
Objectives
The course provides the basic knowledge about complex functions of complex variables, Fourier series, Fourier transform, Laplace transform, partial differential equations.

Knowledge and understanding
At the end of the course the student will know:
- the main notions and results about complex analysis
- the main properties and theorems about Fourier series, Fourier and Laplace transforms
- the methods for the calculus of transforms and inverse transforms
- the proof techniques.

Applying knowledge and understanding
At the end of the course the student will be able to:
- apply theorems and rules in problems solving
- solve exercises of complex analysis
- compute the Fourier series expansion of a function
- calculate Fourier and Laplace transforms and inverse Laplace transforms
- solve ordinary differential equations, systems of ordinary differential equations and integro-differential equations by using transforms
- solve boundary value problems for the Laplace equation, the heat equation and the vibrating string equation
- identify the best methods to efficiently solve a mathematical problem
- explain the resolution of exercises in the written proof
- explain verbally the learned knowledge
- apply the acquired knowledge to different contexts from those presented during the lessons
- deepen the topics dealt by using materials other than the proposed ones.
Prerequisites
Knowledge of integration of functions of one variable, integrals on curves, integrals of differential forms, series expansions, functions of several variables, ordinary differential equations.
Mandatory preparatory teachings: Matematica II
Contents
COMPLEX FUNCTIONS OF COMPLEX VARIABLES (8H LECTURES, 8H EXERCISE). HOLOMORPHIC FUNCTIONS AND THEIR PROPERTIES. THE CAUCHY-RIEMANN CONDITIONS. ELEMENTARY FUNCTIONS. SINGULAR POINTS. CAUCHY’S THEOREM AND CAUCHY’S INTEGRAL FORMULAS. MORERA THEOREM. MEAN VALUE THEOREM. LIOUVILLE THEOREM. TAYLOR’S AND LAURENT SERIES. CLASSIFICATION OF SINGULAR POINTS. RESIDUES AND THE RESIDUE THEOREM.
FOURIER SERIES (6H LECTURES, 6H EXERCISE). EULER-FOURIER COEFFICIENTS. BESSEL INEQUALITY. PUNCTUAL AND UNIFORM CONVERGENCE THEOREMS. SERIES INTEGRATION AND DERIVATION.
FOURIER TRANSFORM (4H LECTURES, 6H EXERCISE). DEFINITION AND PROPERTIES. THE RELATIONSHIP BETWEEN DERIVATION AND MULTIPLICATION BY MONOMIALS. CONVOLUTION TRANSFORM. INVERSION FORMULA.
LAPLACE TRANSFORM (5H LECTURES, 10H EXERCISE). DEFINITION AND PROPERTIES. BEHAVIOR OF LAPLACE TRANSFORM AT INFINITY. INITIAL AND FINAL VALUE THEOREM. LAPLACE TRANSFORMS OF DERIVATIVES. MULTIPLICATION BY POWERS OF T. LAPLACE TRANSFORM OF INTEGRALS. DIVISION BY T. PERIODIC FUNCTIONS. CONVOLUTION TRANSFORM. INVERSE LAPLACE TRANSFORM. INVERSION FORMULAS. APPLICATION TO ORDINARY DIFFERENTIAL EQUATIONS, SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS AND INTEGRO-DIFFERENTIAL EQUATIONS.
PARTIAL DIFFERENTIAL EQUATIONS (2H LECTURES, 5H EXERCISE). INTRODUCTION. HEAT, WAVE AND LAPLACE EQUATIONS. BOUNDARY VALUE PROBLEMS. SOLUTIONS USING SEPARATION OF VARIABLES AND LAPLACE TRANSFORMS.

Teaching Methods
Frontal lectures for a total of 25 hours and classroom exercise sessions for a total of 35 hours.
In order to participate to the final assessment, the student must have attended at least 70% of the hours of assisted teaching activities.
Verification of learning
THE EXAM CONSISTS IN A WRITTEN TEST OF 3 HOURS, PRELIMINARY WITH RESPECT TO THE ORAL EXAMINATION, AND IN AN ORAL INTERVIEW.
IN THE CASE OF A SUFFICIENT WRITTEN PROOF, IT WILL BE EVALUATED THROUGH QUALITATIVE SCALES.
THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY CUM LAUDE), WILL DEPEND ON THE RANGE OF MARKS OF THE WRITTEN TEST, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW.
Texts
MURRAY R. SPIEGEL, VARIABILI COMPLESSE, COLLANA - SCHAUM'S.
MURRAY R. SPIEGEL: ANALISI DI FOURIER, COLLANA - SCHAUM'S.
MURRAY R. SPIEGEL: TRASFORMATE DI FOURIER, COLLANA - SCHAUM'S.
PAUL DUCHATEAU, D. ZACHMANN: PARTIAL DIFFERENTIAL EQUATIONS, SCHAUM’S OUTLINES SERIES. C. D'APICE, R. MANZO: VERSO L’ESAME DI MATEMATICA III, CUES, 2011.
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