1. SylabUZ
2. Faculty of Exact and Natural Sciences
3. winter term 2023/2024
4. Mathematics - Second-cycle studies leading to MS degree
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. 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)