Course detail

Complex Variable Functions

FSI-SKFAcad. year: 2015/2016

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

Czech

Number of ECTS credits

6

Mode of study

Not applicable.

Guarantor

prof. RNDr. Miloslav Druckmüller, CSc.

Department

Institute of Mathematics (ÚM)

Learning outcomes of the course unit

The course provides students with basic knowledge ands skills necessary for using th ecomplex numbers, integrals and residua, usage of Laplace and Fourier transforms.

Prerequisites

Real variable analysis at the basic course level

Co-requisites

Not applicable.

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

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

Course curriculum

Not applicable.

Work placements

Not applicable.

Aims

The aim of the course is to familiarise students with basic properties of complex numbers and complex variable functions.

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

Missed lessons can be compensated for via a written test.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

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

Not applicable.

Classification of course in study plans

  • Programme N3901-2 Master's

    branch M-MAI , 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, 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

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