Complex Variable Functions (FSI-SKF)

Academic year 2023/2024
Supervisor: prof. RNDr. Miloslav Druckmüller, CSc.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:

The aim of the course is to familiarise students with elements of complex analysis and with Fourier transform including applications.  
Learning outcomes and competences:

The course provides students with basic knowledge and skills necessary for using the ecomplex numbers, integrals and residue, usage of  Fourier transforms.
Prerequisites:
Real variable analysis at the basic course level
Course contents:

The aim of the course is to make students familiar with the fundamentals of complex variable functions and Fourier transform.
Teaching methods and criteria:
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes:

Course-unit credit based on a written test.
Exam has a written and an oral part.
Controlled participation in lessons:

Missed lessons can be compensated via a written test.
Type of course unit:
    Lecture  13 × 3 hrs. optionally                  
    Exercise  13 × 2 hrs. compulsory                  
Course curriculum:
    Lecture

1. Complex numbers, Gauss plane, Riemann sphere
2. Functions of complex variable, limit, continuity, elementary functions
3. Derivative, holomorphic functions, harmonic functions, Cauchy-Riemann equations
4. Harmonic functions, geometric interpretation of derivative in complex domain
5. Series and rows of complex functions, power sets, uniform convergence
6. Curves, integral of complex function, primitive function, integral path independence
7. Cauchy's integral formula, uniqueness theorem
8. Taylor and Laurent series
9. Singular points of holomorphic functions, residue, residue theorem
10. Integration by means of residue theory
11. Conformal mapping
12. Fourier transform
13. Fourier transform aplications
    Exercise

1. Complex numbers, Moivre's formula, n-th root
2. Functions of complex variable, limit, continuity, elementary functions
3. Derivative, holomorphic functions, harmonic functions, Cauchy-Riemann equations
4. Harmonic functions, geometric interpretation of derivative in complex domain
5. Series and rows of complex functions, power sets, uniform convergence
6. Curves, integral of complex function, primitive function, integral path independence
7. Cauchy's integral formula, uniqueness theorem
8. Taylor and Laurent series
9. Singular points of holomorphic functions, residue, residue theorem
10. Integration by means of residue theory
11. Conformal mapping
12. Fourier transform
13. Fourier transform aplications
Literature - fundamental:
1. Markushevich A.,I., Silverman R., A.:Theory of Functions of a Complex Variable, AMS Publishing, 2005
2. Šulista M.: Základy analýzy v komplexním oboru. SNTL Praha 1981
3. Druckmüller, M., Ženíšek, A.: Funkce komplexní proměnné, PC-Dir Real, Brno 2000
Literature - recommended:
4. Shanti, N.: Theory of Functions of a Complex Variable , S Chand & Co Ltd 2018
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
N-MAI-P full-time study --- no specialisation -- Cr,Ex 6 Compulsory 2 1 S