1. SylabUZ
2. Faculty of Mathematics, Computer Science and Econometrics
3. winter term 2022/2023
4. WMIiE - oferta ERASMUS - Erasmus programme
5. Real and Complex Analysis

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Real and Complex Analysis - course description

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

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

Scope

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).

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

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.

Recommended reading

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.

Further reading

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

Notes

Modified by dr Dorota Głazowska (last modification: 26-04-2022 18:19)