Functions of Complex Variable
Code | Completion | Credits | Range | Language |
---|---|---|---|---|
01FKO | Z,ZK | 3 | 2+1 | Czech |
- Course guarantor:
- Pavel Šťovíček
- Lecturer:
- Severin Pošta, Pavel Šťovíček
- Tutor:
- Severin Pošta, Pavel Šťovíček
- Supervisor:
- Department of Mathematics
- Synopsis:
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The course starts from outlining the Jordan curve theorem and the Riemann-Stieltjes integral. Then basic results of complex analysis in one variable are explained in detail: the derivative of a complex function and the Cauchy-Riemann equations, holomorphic and analytic functions, the index of a point with respect to a closed curve, Cauchy's integral theorem, Morera's theorem, roots of a holomorphic function, analytic continuation, isolated singularities, the maximum modulus principle, Liouville's theorem, the Cauchy estimates, Laurent series, residue theorem.
- Requirements:
-
The complete introductory course in mathematical analysis on level A or B given at the Faculty of Nuclear Sciences and Physical Engineering
- Syllabus of lectures:
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1. Connected, path-connected, simply connected spaces, the Jordan curve theorem
2. Variation of a function, length of a curve, the Riemann-Stieltjes integral (survey)
3. Derivative of a complex function, the Cauchy-Riemann equations
4. Holomorphic functions, power series, analytic functions
5. Regular curves, integration of a function along a curve (contour integral), the index of a point with respect to a closed curve
6. Cauchy's integral theorem for triangles
7. Cauchy's integral formula for convex sets, relation between holomorphic and analytic functions, Morera's theorem
8. Roots of a analytic function, analytic continuation
9. Isolated singularities
10. The maximum modulus principle, Liouville's theorem
11. The Cauchy estimates, uniform convergence of analytic functions
12. Cauchy's integral theorem (general version)
13. The residue theorem
- Syllabus of tutorials:
- Study Objective:
-
Knowledge: the Jordan curve theorem, construction of the Riemann-Stieltjes integral, basic results of complex analysis in one variable.
Skills: practical usage of complex analysis, applications in evaluation of integrals.
- Study materials:
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Key references:
[1] W. Rudin: Real and Complex Analysis, (McGrew-Hill, Inc., New York, 1974)
Recommended references:
[2] J. B. Conway: Functions of One Complex Variable I, Springer-Verlag, New York, 1978
- Note:
- Time-table for winter semester 2024/2025:
- Time-table is not available yet
- Time-table for summer semester 2024/2025:
- Time-table is not available yet
- The course is a part of the following study plans:
-
- Aplikovaná algebra a analýza (compulsory course in the program)
- Aplikované matematicko-stochastické metody (elective course)
- Fyzikální inženýrství - Inženýrství pevných látek (elective course)
- Jaderná a částicová fyzika (elective course)
- Fyzikální inženýrství - Laserová technika a fotonika (elective course)
- Matematické inženýrství - Matematická fyzika (PS)
- Matematické inženýrství - Matematická informatika (PS)
- Matematické inženýrství - Matematické modelování (PS)
- Kvantové technologie (elective course)
- Quantum Technologies (elective course)
- Nuclear and Particle Physics (elective course)
- Mathematical Engineering - Mathematical Physics (PS)