Academic year 2017/2018 
Supervisor:  prof. RNDr. Miloslav Druckmüller, CSc.  
Supervising institute:  ÚM  
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:  
Courseunit 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.  
Seminars  13 × 2 hrs.  
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, CauchyRiemann 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 

Seminars  1. Complex numbers, sets of complex numbers 2. Functions of complex variable, limit, continuity, elementary functions 3. Derivative, holomorphy functions, harmonic functions, CauchyRiemann 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:  
2. Markushevich A.,I., Silverman R., A.:Theory of Functions of a Complex Variable, AMS Publishing, 2005  
3. Šulista M.: Základy analýzy v komplexním oboru. SNTL Praha 1981  
Literature  recommended:  
1. E.Barvinek, E.Fuchs: Analyticke funkce, , 0 
The study programmes with the given course:  
Programme  Study form  Branch  Spec.  Final classification  Courseunit credits  Obligation  Level  Year  Semester 
M2AP  fulltime study  MMAI Mathematical Engineering    Ac,Ex  6  Compulsory  2  1  S 
Faculty of Mechanical Engineering
Brno University of Technology
Technická 2896/2
616 69 Brno
Czech Republic
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