# Università degli Studi di Napoli "Parthenope"  ## Teaching schedule

2017/2018
Belonging course:
Course of Bachelor's Degree Programme on COMPUTER, BIOMEDICAL AND TELECOMMUNICATION ENGINEERING
Disciplinary sector:
MATHEMATICAL ANALYSIS (MAT/05)
Language:
Italian
Credits:
9
Year of study:
2
Teachers:
Dott. FEO Filomena
Cycle:
First Semester
Hours of front activity:
72

Italian

### Course description

The aim of the course is the knowledge and understanding of basic concepts of complex analysis, Fourier series, Laplace transform, Fourier transform and Distributions.

Learning outcomes (declined compared with respect to the Dublin descriptors)

Knowledge and understanding. Knowledge of the differential and integral calculus for functions of a complex variable, Fourier series, Laplace transform and Fourier Transform. The student will be able to state the basic definitions and to state and prove basic theorems.

Applying knowledge and understanding. The ability to understand the problems proposed during the course, the ability concerning a correct application of the theoretical knowledge. The student will be able to manage complex functions, to solve integration problems, to evaluate Laplace and Fourier transform of a function, to discuss the behavior of Fourier series.

Making judgments. Develop the ability to critically evaluate the problems and propose the most appropriate approach.

Communication skills. Ability to report and present the results with a logical-deductive and synthetic exposition. He must be able to explain (even to non-expert people) the power of some applications of the mathematical tools described in the course, in the field of Engineering.

Ability to learn. Ability to develop, outline, summarize the contents from several sources, in order to achieve a broad overview of the problems connected to the topics discussed in the course. The student will also develop the skill of learning more advances techniques of Mathematical Analysis.

### Prerequisites

The student has to know and to be able to use the tools introduced in Calculus I and II, especially differential calculus and integral calculus and series.

### Syllabus

Preliminaries ( 7 hours of Lectures +1 hour of Exercise session)

Complex functions, topology of complex field, sequences and series.

Complex analysis (12 hours of Lectures + 2 hours of Exercise session)

Cauchy integral theorem, analyticity of holomorphic functions, Laurent series, study of isolated singularities, Residue theorems and application to calculus of real integrals. Different notions of integrability.

Fourier series (8 hours of Lectures + 2 hours of Exercise session)

Definition of Fourier series. Main properties and pointwise convergence. Application to PDE.

Laplace transform (6 hours of lectures+ 2 hours of Exercise session)

Definition of Laplace transform and main properties. Application to ODE.

Fourier transform (6 hours of Lectures + 2 hours of Exercise session)

Definition of Fourier transform and main properties.

Distributions(6 hours of Lectures + 2 hours of Exercise session)

Definition of Distribution, main properties and Fourier transform of a distribution.

Exercises on each topic.

Preliminaries ( 7 hours of Lectures +1 hour of Exercise session)

Complex functions, topology of complex field, sequences and series.

Complex analysis (12 hours of Lectures + 2hours of Exercise session)

Cauchy integral theorem, analyticity of holomorphic functions, Laurent series, study of isolated singularities, Residue theorems and application to calculus of real integrals. Different notions of integrability.

Fourier series (8 hours of Lectures + 2 hours of Exercise session)

Definition of Fourier series. Main properties and pointwise convergence. Application to PDE.

Laplace transform (6 hours of lectures+ 2 hours of Exercise session)

Definition of Laplace transform and main properties. Application to ODE.

Fourier transform (6 hours of Lectures + 2 hours of Exercise session)

Definition of Fourier transform and main properties.

Distributions(6 hours of Lectures + 2 hours of Exercise session)

Definition of Distribution, main properties and Fourier transform of a distribution.

### Teaching Methods

Lectures, exercise sessions and homeworks.

### Textbooks

English Books:

Complex Variables and the Laplace Transform for Engineers by Wilbur R.
Lepage

Italian books:

S. Abenda - S. Matarasso, Metodi Matematici, Esculapio.

G.C. Barozzi, Matematica per l`Ingegneria dell`Informazione, Zanichelli.

M. Codegone, Metodi Matematici per l`Ingegneria, Zanichelli.

### Learning assessment

The final exam tests the achievement of the training objectives.

The final examination consists in a discussion. The student has to solve some easy exercises to verify the knowledge and understanding of arguments. Moreover a discussion about some connections and comparison between different considered arguments.