FUNCTIONS OF A COMPLEX VARIABLE
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- Versione italiana
- Academic year
- 2019/2020
- Teacher
- CINZIA BISI
- Credits
- 6
- Didactic period
- Primo Semestre
- SSD
- MAT/03
Training objectives
- This is the first course on the theory of functions in Complex Analysis.
The main goal of the course consists in providing the basic concepts ans results of the theory of complex functions of one variable.
The main acquired knowledge will be:
basic knowledge of a complex function , also of multi-valued functions;
basic knowledge to tackle the study of complex functions, sequences and series of functions, series representations, integrals of continuous complex functions along varios types of paths of integration;
basic knowledge of analytic function and their power series expansions.
The basic acquired abilities will be:
analysis of the behavior of complex functions;
study of the analitic and algebraic properties of functions of a complex variable and the geometric aspects of the mapping defined by them;
use of the properties of complex functions to solve some problems in the real field. Prerequisites
- We assume the student to be familiar with the real and complex number system, as well as with the properties of the real functions of one and several real variables , as developed in the usual courses in calculus and in real variables.
We assume also some acquaintance with the fundamental concepts and propositions of abstract algebra, modern geometry and topology. Course programme
- The course forecasts 48 hours of frontal lectures. Functions of one complex variable: basic definitions, elmentary functions; analytic functions; equations of Cauchy-Riemann; sequences and series of complex functions; convergences; power series; conformal mappings; elementary transcendental functions; complex integration; Cauchy's theorem for star regions and integral formula for the disc; properties of analytic functions: the Taylor series; Liouville's theorem, Riemann identity theorem, Riemann prolongation theorem; the open mapping theorem; isolated singularities of complex functions, meromorphic functions; Cauchy's theorem for the annulus; Laurent's theorem; the expansion in Laurent series for holomorphic functions in the annulus; Index of a path towards a point; regions homologically simple connected; general Cauchy's theorem; the calculus of residues:the residue theorem and its applications in the evaluation of real integrals.
Didactic methods
- Frontal lectures on all the course's topics : written exposition of the theory with performed exercises carried out as illustrative examples of the treated theory.
Learning assessment procedures
- The aim of the exam is to verify at which level the learning objectives previously described have been acquired.
The examination is divided in 2 sections that will be place in different days.
One written test with solutions of numerical exercises based on all topics tackled in the class. The aim of the test is evaluating how deeply the student has studied the subject and how he is able to understanding the basic topics analyzed; the section is selective as the student who does not show a sufficient knowledge of the subjects, cannot be admitted to the oral section. The time allowed for the written test is 1 hour and 30 minutes . To pass the test it is rcquired to get at least 18 points out of 30 ; It is only allowed to consult personally handwritten notes, written on an A4 sheet that can hold all the knowledge deemed useful for passing the test ;
one oral section where the knowledge of the basic concepts and results of the theory of complex functions and the ability of linking different subjects related to complex functions are evaluated, rather than the ability of “repeating” specific topics tackled in the cours; demonstrations of some of the main theorems are required;
The final mark takes account of the mark of the written test and the oral presentation that does show a sufficient knowledge of the subjects; if this knowledge is not considered sufficient, it is necessary to repeat only the oral section in a subsequent exam session.
To pass the exam it is necessary to get at least 18 points of 30. Reference texts
- Specific topics can be further developed in the following tests
A.Del Centina - Teoria delle funzioni di una variabile complessa - Aracne editrice
M.O.Gonzalez - Classical complex Analysis - Dekker
