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

Language of instruction

Czech

Number of ECTS credits

6

Supervisor

prof. RNDr. Miloslav Druckmüller, CSc.

Department

Institute of Mathematics

Entry knowledge

Real variable analysis at the basic course level

Rules for evaluation and completion of the course

Course-unit credit based on a written test.
Exam has a written and an oral part.


Missed lessons can be compensated via a written test.

Aims

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


The course provides students with basic knowledge and skills necessary for using the ecomplex numbers, integrals and residue, usage of  Fourier transforms.

The study programmes with the given course

Programme N-MAI-P: Mathematical Engineering, Master's, 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