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Lecturer:
Elena Bonetti
Course name: Metodi matematici
Course code: 500541
Degree course: Ingegneria Meccatronica
Disciplinary field of science: MAT/05
L'insegnamento costituisce attività di base per: Ingegneria Meccatronica
University credits: CFU 6
Course website: http://www.imati.cnr.it/~gianazza/metodi.html
Specific course objectives
The course aims to introduce some of the main methods of mathematical analysis, and to provide students with useful operative tools for applications to signal theory and optimization problems. Main objectives are: i) lead the students to easily use the most important functions of complex variable and provide the basic knowledge of the related theory, ii) introduce the concept of convergence of sequences and series of functions and present the fundamental results about Fourier series, Fourier transform and Laplace transform iii) illustrate some techniques and applications of these transformations to simple differential problems.
Course programme
Introduction to complex analysis
- Complex numbers
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Power series in complex fields: radius of convergence and formula for its calculation
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Exponential function, trigonometric functions, roots and logarithm of complex numbers
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Complex derivative and holomorphic functions, holomophisms of power series
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Path integrals in complex field
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Cauchy theorem, analyticity of holomorphic functions
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Singularity and Laurent series, residue Theorem
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Application to the calculation of integrals, Jordan Lemma.
Fourier series
- Periodic signals, trigonometric polynomials, Fourier series, comparison between trigonometric form and exponential form
- The best approximation problem and the convergence in energy sense
- Parseval identity and application to the calculation of numeric series
- Pointwise and uniform convergence, application to the calculation of numeric series Fourier transform for integrable functions
- Definition of Fourier transform, fundamental properties, link with Fourier series
- Riemann-Lebesgue lemma, examples of calculation of Fourier transforms
- Fourier transform of signal with finite energy and the Plancherel Theorem
- The inversion theorem
Laplace transform
- Definition, main properties and examples of calculation
- Link with Fourier transform
- Inversion of Laplace transform, Heaviside formula.
- Application to linear differential equations with constant coefficients and to the Cauchy problem
Convolution
- Definition and main properties, examples
- Link with Fourier and Laplace transformation
Course entry requirements
Course structure and teaching
Lectures (hours/year in lecture theatre): 38
Practical class (hours/year in lecture theatre): 14
Practicals / Workshops (hours/year in lecture theatre): 0
Project work (hours/year in lecture theatre): 0
Suggested reading materials
M. Codegone. Metodi Matematici per l'Ingegneria. Zanichelli.
F. Tomarelli. Esercizi di Metodi Matematici per l'Ingegneria. CLU.
Testing and exams
Solving exercises and interrogation
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