Course detail
Complex Variable Functions
FSI-SKF Acad. year: 2024/2025 Summer semester
The aim of the course is to make students familiar with the fundamentals of complex variable functions and Fourier transform.
Language of instruction
Czech
Number of ECTS credits
6
Supervisor
Department
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, 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
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. Conformal mapping
12. Fourier transform
13. Fourier transform aplications