1. SylabUZ
2. Wydział Nauk Ścisłych i Przyrodniczych
3. semestr zimowy 2024/2025
4. WMIiE - oferta ERASMUS - Program Erasmus
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. Caratheodory Theorem and a construction of Lebesgue measure  (2h).

2. Measurable functions Theorems of Jegorov and Lusin (2h).

3. Integral with respect to a measure. Integrable functions. Theorems of Fubini and Radon-Nikodym (4h).

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

Exercises
I. MEASURE THEORY
1. Caratheodory theorem and a construction of  Lebesgue measure (1h).

2. Measurable sets, measurable functions and Thorems of Jegorov, Lusin (2h).
3. Integral with respect to a measure. Integrable functions. Theorems of Fubini and Radon-Nikodym (3h).

II. THEORY OF COMPLEX FUNCTIONS
1. Complex derivative, Cauchy-Riemann equations, analytic (holomorphic) function (4h).
2. Curve integral of a complex function, Cauchy integral theorem, Cauchy’s integral formula ( 6h).
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 (5h).
5. Theorem of residues and their applications, meromorphic functions (4h).

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

1. Examination of the students’ preparation and their activity during exercises.
2. Tests, of different level of difficulty, permitting to verify the level of student commanding of the particular effects of education.
3. Exam (written and oral) checks the understanding of the basic notions, knowledge of the important examples and the proofs of some chosen theorems.
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. Franciszek Leja, Funkcje zespolone, Biblioteka Matematyczna, PWN, 1973; Rozdziały VII-IX.
2. Walter Rudin, Real and Complex Analysis, Third Edition, Mc Graw - Hill Company, 1987.

Literatura uzupełniająca

1. Roman Sikorski, Funkcje rzeczywiste I, Państwowe Wydawnictwo Naukowe, Warszawa 1957.
2. W. Kołodziej, Analiza matematyczna, Państwowe Wydawnictwo Naukowe, Warszawa

Uwagi

Zmodyfikowane przez dr Dorota Głazowska (ostatnia modyfikacja: 18-04-2024 13:08)