Complex Variable Functions (FSI-SKF)

Academic year 2025/2026
Supervisor: prof. RNDr. Miloslav Druckmüller, CSc.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:
 
Learning outcomes and competences:
 
Prerequisites:
 
Course contents:

The aim of the course is to make students familiar with the fundamentals of complex variable functions
Teaching methods and criteria:
 
Assesment methods and criteria linked to learning outcomes:
 
Controlled participation in lessons:
 
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 
8. Taylor series, uniqueness theorem                                                            9.  Laurent series                                                                                          10. Singular points of holomorphic functions, residue, residue theorem
11. Integration by means of residue theory
12. Real integrals by means of residue theory
13. Conformal mapping
    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. Integration by means of residue theory
12. Real integrals by means of residue theory
13. Conformal mapping
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