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 to find the sum of certain numerical series, to compute the Fourier or the Laplace trasforms of functions, to solve systems of linear differential equations, integral equations, integro-differential equations and simple partial differential equations by means of Fourier or Laplace transforms, as well as to operate with 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 or simple partial differential equations. The chapter on distributions will introduce the student to the modern way of studying 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, to develop a function in trigonometric  series, to solve a system of linear ODEs and simple PDEs, as well as to work with distributions. 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 (three hours and five exercises) and an oral test. Both may also be carried out on-line, shouldthe conditions require it. To ensure equal opportunities and in compliance with current laws, interested students may request a personal interview in order to plan any compensatory and/or dispensatory measures based on educational objectives and specific needs. Students can also contact the CInAP (Centro per l’integrazione Attiva e Partecipa ta — Servizi per le Disabilità e/o i DSA) referring teacher within the DMI (https://www.cinap.unict.it/content/referenti)

1. Periodic, piecewise continuous and piecewise regular functions. Fourier expansions. Pointwise and uniform convergence of Fourier series, integration term by term. Calculating the sum of convergent numerical series. Fourier's series in Hilbert spaces.

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

3. Fourier transformation. Definition and basic properties, Riemann-Lebesgue's theorem. 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. Derivatives and transforms, the final value theorem. Convolutions and their tranformations. tinversion formulae. Applications to linear differential equations and systems with constant coefficients.

5. Outline of distributions. The test function space. Distributions. The space L1loc(R). Distribution functions. The Dirac distribution. Sequences of distributions. Operations. Derivative of a distribution. Meaningful special cases. Fourier transform of a tempered distribution. Basic properties. Laplacce transform of a tempered distribution.

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.

4) R.E. GREENE - S.G. Krantz, Function Theory of One Complex Variable, AMS, Providence , 2006.

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.

1) Prove one of the following fundamental results: Cauchy-Riemann holomorphy conditions; Cauchy-Goursat theorem; Cauchy integral formulas; Liouville's theorem; Hermite's theorem; fundamental theorem of algebra; residue theorem.
2) Discuss the pointwise and uniform convergence of the Fourier series.
3) Define the Fourier transform and prove that it is infinitesimal at infinity. Riemann-Lebesgue theorem.
4) Define the Laplace transform and prove the formula for the transformation of a periodic function.
5) Explain the concept of distribution, prove some formulas for the derivative of a distribution, and discuss sequences of distributions.
6) Define the Fourier or Laplace transform of a tempered distribution and prove some basic formulas.