MATHEMATICAL ANALYSIS III

The aim of the course of Analisi Matematica III is the study of the theory of functions of a complex variable and the Integral Transforms. The student will also develop the ability to apply the concepts learned to the resolution of problems and non-trivial exercises.

In particular, the learning objectives of the course, according to the Dublin descriptors, are:

1. Knowledge and understanding: The student will learn some concepts of Complex
   Analysis and will develop both computing ability and the capacity of manipulating some common
   mathematical structures, as functions of a complex variable and the Integral Transforms.
2. Applying knowledge and understanding: The student will be able to apply the acquired
   knowledge in the basic processes of mathematical modeling of classical problems arising from
   Engineering.
3. Making judgements: The student will be stimulated to autonomously deepen his/her knowledge
   and to carry out exercises on the topics covered by the course. Constructive discussion between
   students and constant discussion with the teacher will be strongly recommended so that the
   student will be able to critically monitor his/her own learning process.
4. Communication skills: The frequency of the lessons and the reading of the recommended books
   will help the student to be familiar with the rigor of the mathematical language. Through constant
   interaction with the teacher, the student will learn to communicate the acquired knowledge with
   rigor and clarity, both in oral and written form. At the end of the course the student will have
   learned that mathematical language is useful for communicating clearly in the scientific field.
5. Learning skills: The student will be guided in the process of perfecting his/her study method. In
   particular, through suitable guided exercises, he/she will be able to independently tackle new
   topics, recognizing the necessary prerequisites to understand them.

The course consists of blackboard lessons on the theoretical parts and subsequent problem sessions. Occasionaly, electronic devices might be used.

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the program planned and outlined in the Syllabus.

Complex numbers and the complex plane. Functions on the complex plane. Continuous functions. Holomorphic functions. Integration along curves. Goursat’s theorem. Cauchy’s integral formulas. Morera’s theorem. Power series. Analyticity of power series. Laurent series. Singular points. Laurent expansions and the residue theorem. Residue calculus. Zeros and poles. The Fourier Transform. The Laplace Transform. The Zeta Transform. Distributions. Limits of Distributions. The Fourier Transform of a Tempered Distribution.

Di Fazio G., Frasca M. Metodi Matematici per l’Ingegneria,, Monduzzi Editoriale.