COMPLEX ANALYSIS AND INTEGRAL TRANSFORMS

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.

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.

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.

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.