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Lecturer(s)
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Brzezina Miroslav, doc. RNDr. CSc., dr. h. c.
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Course content
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1. Basic set concepts. Set of complex numbers, absolute value, algebraic and trigonometric form. 2. Moivre's theorem and its consequences, binomial equation. 3. Display of complex numbers in a plane, construction of images of arithmetic operations. 4. Circular inversion, definition, description of circular inversion using complex numbers, image of line and circle in circular inversion. 5. Analytical geometry in a complex equation. The introduction of infinity, the complex sphere and its topology. 6. Limit of a complex sequence, basic properties. 7. Complex functions of one complex variable, operations with functions. Continuity and its properties. 8. The limit of a complex function and its properties. 9. Definition of derivation, basic theorems. 10. Cauchy-Riemann theorem. 11. Number series in a complex field. Convergence and absolute convergence of series. 12. Power series in a complex field. Radius of convergence, derivation of power series. Taylor series. 13. Introduction of functions sin, cos, exp, their properties. Calculating values and solving simple equations. Polynomial logarithm. 14. Laurent series.
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Learning activities and teaching methods
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Monological explanation (lecture, presentation,briefing), Written assignment presentation and defence
- Class attendance
- 84 hours per semester
- Preparation for exam
- 155 hours per semester
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Learning outcomes
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The set of complex numbers, the topology of the complex sphere. Basic analysis of complex functions, Cauchy-Riemann theorem. Complex series, Taylor series, introduction of elementary functions. Laurent series.
Knowlige of fundamentals of function theory.
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Prerequisites
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Passing of mathematical lectures of first four semestrs.
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Assessment methods and criteria
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Oral exam, Written exam
Credit: Working out a semestral work. Exam: Written.
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Recommended literature
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Černý, I. Úvod do analýzy v komplexním oboru..
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