Functions of Complex Variable

Garant předmětu:

Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

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:

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:

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:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• BS Matematické inženýrství - Matematické modelování (compulsory course of the specialization)
• Aplikovaná algebra a analýza (compulsory course in the program)
• Aplikované matematicko-stochastické metody (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)