Course detail

Complex Variable Functions

FSI-SKF-AAcad. year: 2024/2025

The aim of the course is to make studetns familiar with the fundamentals of complex variable functions. The course focuses on the following areas: complex numbers, elementar functions of complex variable, holomorfous functions, derivative and integral of complex variable functions, meromorphous functions, Taylor and Laurent series, residua, residua theorem and its applications in integral computing, conformous mapping, homography and other examples of usage of conformous mapping, Laplace transform and its basic properties, Dirac and delta functions and its applications in differential equations solution, Fourier transform.

Language of instruction

English

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 for via a written test.

Aims

The aim of the course is to familiarise students with basic properties of complex numbers and complex variable functions.
The course provides students with basic knowledge ands skills necessary for using th ecomplex numbers, integrals and residua, usage of Laplace and 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
Markushevich A.,I., Silverman R., A.:Theory of Functions of a Complex Variable, AMS Publishing, 2005
Šulista M.: Základy analýzy v komplexním oboru. SNTL Praha 1981

Recommended reading

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

Classification of course in study plans

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

Type of course unit

 

Lecture

39 hod., optionally

Teacher / Lecturer

prof. RNDr. Miloslav Druckmüller, CSc.

Syllabus

1. Complex numbers, sets of complex numbers
2. Functions of complex variable, limit, continuity, elementary functions
3. Derivative, holomorphy functions, harmonic functions, Cauchy-Riemann equations
4. Harmonic functions, geometric interpertation of derivative
5. Series and rows of complex functions, power sets
6. Integral of complex function
7. Curves
8. Cauchy's theorem, Cauchy's integral formula
9. Theorem about uniqueness of holomorphy functions
10. Isolated singular points of holomorphy functions, Laurent series
11. Residua
12. Conformous mapping
13. Fourier transform

Exercise

26 hod., compulsory

Teacher / Lecturer

prof. RNDr. Miloslav Druckmüller, CSc.

Syllabus

1. Complex numbers, sets of complex numbers
2. Functions of complex variable, limit, continuity, elementary functions
3. Derivative, holomorphy functions, harmonic functions, Cauchy-Riemann equations
4. Harmonic functions, geometric interpertation of derivative
5. Series and rows of complex functions, power sets
6. Integral of complex function
7. Curves
8. Cauchy's theorem, Cauchy's integral formula, Liouville's theorem
9. Theorem about uniqueness of holomorphy functions
10. Isolated singular points of holomorphy functions, Laurent series
11. Residua
12. Conformous mapping
13. Laplace transform