Mathematics for Cybernetics
Code  Completion  Credits  Range  Language 

A3M01MKI  Z,ZK  8  4P+2S  Czech 
 Lecturer:
 Tutor:
 Supervisor:
 Department of Mathematics
 Synopsis:

The goal is to explain basic principles of complex analysis and its applications. Fourier transform, Laplace transform and Ztransform are treated in complex field. Finally random processes (stacinary, markovian, spectral density) are treated.
 Requirements:
 Syllabus of lectures:

1. Complex plane. Functions of compex variables. Elementary functions.
2. CauchyRiemann conditions. Holomorphy.
3. Curve integral. Cauchy theorem and Cauchy integral formula.
4. Expanding a function into power series. Laurent series.
5. Expanding a function into Laurent series.
6. Resudie. Residue therorem.
7. Fourier transform.
8. Laplace transform. Computing the inverse trasform by residue method.
9. Ztransform and its applications.
10. Continuous random processes and time series  autocovariance, stacionarity.
11. Basic examples  Poisson processes, gaussian processes, Wiener proces, white noice.
12. Spectral density of the stacionary process and its expression by means of Fourier transform. Spectral decomposition of moving averages.
13. Markov chains with continuous time and general state space.
 Syllabus of tutorials:

1. Complex plane. Functions of compex variables. Elementary functions.
2. CauchyRiemann conditions. Holomorphy.
3. Curve integral. Cauchy theorem and Cauchy integral formula.
4. Expanding a function into power series. Laurent series.
5. Expanding a function into Laurent series.
6. Resudie. Residue theroem
7. Fourier transform
8. Laplace transform. Computing the inverse trasform by residue method.
9. Ztransform and its applications.
10. Continuous random processes and time series  autocovariance, stacionarity.
11. Basic examples  Poisson processes, gaussian processes, Wiener proces, white noice.
12. Spectral density of the stacionary process and its expression by means of Fourier transform. Spectral decomposition of moving averages.
13. Markov chains with continuous time and general state space.
 Study Objective:
 Study materials:

[1] S.Lang. Complex Analysis, Springer, 1993.
[2] L.Debnath: Integral Transforms and Their Applications, 1995, CRC Press, Inc.
[3] Joel L. Shiff: The Laplace Transform, Theory and Applications, 1999, Springer Verlag.
 Note:
 Further information:
 http://math.feld.cvut.cz/veronika/vyuka/b3b01kat.htm
 No timetable has been prepared for this course
 The course is a part of the following study plans: