Course detail

Complex Variable Functions

FSI-SKFAcad. year: 2024/2025

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

Mode of study

Not applicable.

Guarantor

prof. RNDr. Miloslav Druckmüller, CSc.

Department

Institute of Mathematics (ÚM)

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.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Druckmüller, M., Ženíšek, A.: Funkce komplexní proměnné, PC-Dir Real, Brno 2000 (CS)
Markushevich A.,I., Silverman R., A.:Theory of Functions of a Complex Variable, AMS Publishing, 2005 (EN)
Šulista M.: Základy analýzy v komplexním oboru. SNTL Praha 1981 (CS)

Recommended reading

Shanti, N.: Theory of Functions of a Complex Variable , S Chand & Co Ltd 2018 (EN)

Classification of course in study plans

  • Programme N-MAI-P Master's 1 year of study, summer semester, compulsory

  • Programme C-AKR-P Lifelong learning

    specialization CLS , 1 year of study, summer semester, elective

Type of course unit

 

Lecture

39 hod., optionally

Teacher / Lecturer

prof. RNDr. Miloslav Druckmüller, CSc.

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 hod., compulsory

Teacher / Lecturer

prof. RNDr. Miloslav Druckmüller, CSc.

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