Complex Variable Functions (FSI-SKF)

Academic year 2018/2019
Supervisor: prof. RNDr. Miloslav Druckmüller, CSc.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:
The aim of the course is to familiarise students with basic properties of complex numbers and complex variable functions.
Learning outcomes and competences:
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
Course contents:
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.
Teaching methods and criteria:
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.
Controlled participation in lessons:
Missed lessons can be compensated for via a written test.
Type of course unit:
    Lecture  13 × 3 hrs. optionally                  
    Exercise  13 × 2 hrs. compulsory                  
Course curriculum:
    Lecture 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 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
Literature - fundamental:
1. Markushevich A.,I., Silverman R., A.:Theory of Functions of a Complex Variable, AMS Publishing, 2005
2. Šulista M.: Základy analýzy v komplexním oboru. SNTL Praha 1981
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
M2A-P full-time study M-MAI Mathematical Engineering -- Cr,Ex 6 Compulsory 2 1 S