Course detail

# Complex Variable Functions

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.

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.

Recommended optional programme components

Not applicable.

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)

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.

Language of instruction

Czech

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.

Classification of course in study plans

• Programme M2A-P Master's

branch M-MAI , 1. year of study, summer semester, 5 credits, compulsory

#### Type of course unit

Lecture

39 hours, optionally

Teacher / Lecturer

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 hours, compulsory

Teacher / Lecturer

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

eLearning

eLearning: opened course