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# Course detail

## Complex Variable Functions

Course unit code: FSI-SKF
Type of course unit: compulsory
Level of course unit: Master's (2nd cycle)
Year of study: 1
Semester: summer
Number of ECTS credits:
 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.
 Mode of delivery: 90 % face-to-face, 10 % distance learning
 Prerequisites: Real variable analysis at the basic course level
 Co-requisites: Not applicable.
 Recommended optional programme components: Not applicable.
 Course contents (annotation): 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.
 Recommended or required reading: E.Barvinek, E.Fuchs: Analyticke funkce, , 0Markushevich 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
 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.
 Course curriculum: 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.

Type of course unit:

Lecture: 39 hours, optionally prof. RNDr. Miloslav Druckmüller, CSc. 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 26 hours, compulsory prof. RNDr. Miloslav Druckmüller, CSc. 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

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