Functions of Complex Variable
Code  Completion  Credits  Range  Language 

01FKO  Z,ZK  3  2+1  Czech 
 Lecturer:
 Pavel Šťovíček (guarantor)
 Tutor:
 Pavel Šťovíček (guarantor)
 Supervisor:
 Department of Mathematics
 Synopsis:

The course starts from outlining the Jordan curve theorem and the RiemannStieltjes integral. Then basic results of complex analysis in one variable are explained in detail: the derivative of a complex function and the CauchyRiemann 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:

1. Connected, pathconnected, simply connected spaces, the Jordan curve theorem
2. Variation of a function, length of a curve, the RiemannStieltjes integral (survey)
3. Derivative of a complex function, the CauchyRiemann 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 RiemannStieltjes integral, basic results of complex analysis in one variable.
Skills: practical usage of complex analysis, applications in evaluation of integrals.
 Study materials:

Key references:
[1] W. Rudin: Real and Complex Analysis, (McGrewHill, Inc., New York, 1974)
Recommended references:
[2] J. B. Conway: Functions of One Complex Variable I, SpringerVerlag, New York, 1978
 Note:
 Timetable for winter semester 2022/2023:
 Timetable is not available yet
 Timetable for summer semester 2022/2023:
 Timetable is not available yet
 The course is a part of the following study plans:

 BS Matematické inženýrství  Matematické modelování (compulsory course of the specialization)
 BS Matematická informatika (compulsory course of the specialization)