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Real and Complex Analysis - course description

General information
Course name Real and Complex Analysis
Course ID 11.1-WK-MATED-RCA-S22
Faculty Faculty of Exact and Natural Sciences
Field of study Mathematics
Education profile academic
Level of studies Second-cycle studies leading to MS degree
Beginning semester winter term 2023/2024
Course information
Semester 1
ECTS credits to win 7
Course type obligatory
Teaching language english
Author of syllabus
  • prof. dr hab. Witold Jarczyk
  • prof. dr hab. Janusz Matkowski
Classes forms
The class form Hours per semester (full-time) Hours per week (full-time) Hours per semester (part-time) Hours per week (part-time) Form of assignment
Lecture 30 2 - - Exam
Class 30 2 - - Credit with grade

Aim of the course

The aim is to improve the acquaitance of a student of deeper facts in real analysis and give him opportunity to gain the standard knowledge in the theory of complex functions in single variable.

Prerequisites

Average education in the basic notions and results in real analysis.

Scope

Lecture
I. MEASURE THEORY
1. Theorems of Jegorov, Lusin (4 h.).
2. Theorems of Fubini and Radon-Nikodym (4 h.).
II. THEORY OF COMPLEX FUNCTIONS
1. Complex derivative, Cauchy-Riemann equations, analytic (holomorphic) function (4 h.).
2. Curve integral of a complex function, Cauchy integral theorem, Cauchy’s integral formula ( 4 h.).
3. Expansion of an analytic function in power series, entire functions, theorem of Liouville, maximum principle, Schwarz lemma (5 h.).
4. Laurent series, singular points and their classification, residuum (5 h.).
5. Theorem of residues and their applications, meromorphic functions (4 h.).

Exercises
I. MEASURE THEORY
1. Thorems of Jegorov, Lusin (3 h.)
2. Theorems of Fubini and Radon-Nikodym (3 h.)

II. THEORY OF COMPLEX FUNCTIONS
1. Complex derivative, Cauchy-Riemann equations, analytic (holomorphic) function (4 h.).
2. Curve integral of a complex function, Cauchy integral theorem, Cauchy’s integral formula ( 6 h.).
3. Expansion of an analytic function in power series, entire functions, theorem of Liouville, maximum principle, Schwarz lemma (5 h.).
4. Laurent series, singular points and their classification, residuum (5 h.).
5. Theorem of residues and their applications, meromorphic functions (4 h.).

Teaching methods

Conventional lecture; problem lecture
Auditorium exercises – solving standard problems enlightening the significance of the theory, exercises on applications, solving problems.

Learning outcomes and methods of theirs verification

Outcome description Outcome symbols Methods of verification The class form

Assignment conditions

Passing the exam: the weighted mean of notes of exercises (40%) and the exam (60%).
A positive note of the exercises is the necessary condition to be admitted to the exam. A positive note of the exam attests the subject.

Recommended reading

  1. W. Rudin, Real and Complex Analysis, Mc Graw - Hill Company, 1987.                                                                                                                
  2. R.H. Dyer, D.E. Edmunds, From Real to Complex Analysis, Springer, 2014.                                                                                                        
  3. Rajnikant Sinha, Real and Complex Analysis, Springer, 2018.
     

Further reading

1. B.R. Gelbaum, Problems in Real and Complex Analysis, Springer, 1992.
 

 

Notes


Modified by dr Ewa Sylwestrzak-Maślanka (last modification: 10-04-2024 15:46)