Course detail

Complex Variable Functions

FSI-SKF Acad. year: 2025/2026 Summer semester

The aim of the course is to make students familiar with the fundamentals of complex variable functions

Supervisor

prof. RNDr. Miloslav Druckmüller, CSc.

Department

Institute of Mathematics

Learning outcomes of the course unit

Prerequisites

Planned learning activities and teaching methods

Assesment methods and criteria linked to learning outcomes

Language of instruction

Czech

Aims

Specification of controlled education, way of implementation and compensation for absences

The study programmes with the given course

Programme N-MAI-P: Mathematical Engineering, Master's
branch ---: no specialisation, 6 credits, compulsory

Type of course unit

 

Lecture

39 hours, optionally

Syllabus

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

26 hours, compulsory

Syllabus

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