# COMPLEX ANALYSIS AND INTEGRAL TRANSFORMS

Academic Year 2023/2024 - Teacher: Salvatore Angelo MARANO

## Expected Learning Outcomes

The student will acquire the ability to calculate definite integrals, not computable in a simple way, by using the residue theorem, to develop in Fourier series periodic functions and find the sum of certain numerical series, to compute the Fourier and Laplace trasforms of functions, to systems of linear differential equations, integral equations, and integro-differential equations by means of Fourier or Laplace transforms, as well as operate with simple distributions.

In particular, the course has the following objectives:

Knowledge and understanding: Basic complex analysis and topics of the Fourier series will be treated, even in order to deepen and unify certain concepts and methods learned in previous courses of mathematical analysis. The section on Fourier and Laplace trasforms will provide students with the theoretical knowledge needed to apply these tools to important problems, such as linear ordinary differential equations.

Applying knowledge and understanding: The student will learn to solve definite, generalized, or improper integrals, not elementarily computable, with the residue method. He will be able to study the developability and find the development in Fourier series of periodic functions, and calculate the sum of certain numerical series, and he will learn to apply the Fourier and Laplace trasforms to important practical problems.

Making judgments: At the end of the course, students will be able to find the most suitable mathematical tool to calculate a given integral, develop a function in Fourier series, and solve a system of ordinary differential equations. They will be also able to judge which of the basic analysis concepts can naturally be extended to the complex analysis framework.

Communication skills: During the lessons, students will be constantly invited to speak, expressing their point of view, both on theoretical topics and applications. This aims to develop their critical sense and intuition, as well as to get them used to communicate with a mathematically correct language.

Learning skills: They will be stimulated and periodically checked with classroom exercises and simple theoretical topics to be developed individually.

## Course Structure

Lectures and exercises in the classroom.

Verification of learning involves a written test and an oral test. Both can also be carried out electronically, if conditions will require it.

PLEASE NOTE: Information for students with disabilities and / or DSA

To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or dispensatory measures, based on the didactic objectives and specific needs.

It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and / or SLD) of our Department, prof. Filippo Stanco.

## Required Prerequisites

Having passed the Mathematical Analysis II exam.

## Attendance of Lessons

strongly recommended

## Detailed Course Content

1. Periodic, piecewise continuous and regular functions. Developments in Fourier series. Pointwise and uniform convergence of Fourier series, integration term by term. Calculating the sums of convergent numerical series.

2. Derivation and integration in the complex field. Cauchy formula, Liouville theorem, proof of the fundamental theorem of algebra. Theorem of Hermite. Laurent theorem on the developability in two-sided power series. Isolated singular points, classification and characterization. Calculation of residues in the poles, the residue theorem and its applications.

3. Fourier transformation. Definition and basic properties. Transforms of the functions rect (x), exp (-ax^2) and exp (-a | x |) with a> 0, 1 / (1 + x^2). Derivative and transforms. Convolutions and their transforms. Inversion formulas.

4. Laplace transformation. Definition and basic properties. Transforms the functions H (t), sin (ωt), cos (ωt), [t]. Transform of periodic functions. Derivative and transformed, the final value theorem. Convolutions and their transformed. inversion formula. Applications to linear differential equations and systems with constant coefficients.

5. Outline of distributions. The test function space. Distributions. The L^1_loc (R) space. Distribution functions. The Dirac distribution. Sequences of distributions. Operations. Derivative of a distribution. Significant special cases. Tempered distributions and Fourier transforms.

## Textbook Information

1) G. C. BAROZZI, Matematica per l’Ingegneria dell’Informazione, Zanichelli, Bologna, 2003.

2) M. BRAMANTI, Metodi di Analisi Matematica per l'Ingegneria, Società Editrice Esculapio, Bologna, 2019.

3) G. DI FAZIO - M. FRASCA, Metodi Matematici per l’Ingegneria, Monduzzi, Bologna, 2003.

## Course Planning

SubjectsText References
1Fourier series.1)-3)
2The residue theorem1)-3)
3Fourier and Laplace transforms1)-3)
4Spaces of test functions. Distributions and tempered distributions.1)-3)

## Learning Assessment

### Learning Assessment Procedures

During the course, there will be two written tests in progress, one in the middle of the course and one at the end. Students who pass both are exempted from taking the complete written test required for each session. After the written test, an oral interview must be taken.

Both for the ongoing tests and for the final exam, the following will be taken into account: clarity of presentation, completeness of knowledge, ability to connect different topics. The student must demonstrate that he has acquired sufficient knowledge of the main topics covered during the course, and that he is able to carry out at least the simplest of the assigned exercises. There is no average between the written and oral grades. The following criteria will normally be followed to assign the grade:
not approved: the student has not acquired the basic concepts and is not able to carry out the exercises.
18-23: the student demonstrates minimal mastery of the basic concepts, his skills in exposition and connection of contents are modest, he is able to solve simple exercises. 24-27: the student demonstrates good mastery of the course contents, his presentation and content connection skills are good, he solves the exercises with few errors.
28-30 cum laude: the student has acquired all the contents of the course and is able to explain them fully and connect them with a critical spirit; she solves the exercises completely and without errors.
Verification of learning can also be carried out electronically, should the conditions require it.

See Studium.