1. SylabUZ
2. Wydział Nauk Ścisłych i Przyrodniczych
3. semestr zimowy 2023/2024
4. Mathematics - drugiego stopnia z tyt. magistra
5. Real and Complex Analysis

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Real and Complex Analysis - opis przedmiotu

Cel przedmiotu

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.

Wymagania wstępne

Zapisz zmiany

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

Zakres tematyczny

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.).

Metody kształcenia

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

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

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.

Literatura podstawowa

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.
    

Literatura uzupełniająca

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

 

Uwagi

Zmodyfikowane przez dr Ewa Sylwestrzak-Maślanka (ostatnia modyfikacja: 10-04-2024 15:46)