Subject description - AE3M01MKI
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Explanatory Notes
Instructions
Anotation:
The goal is to explain basic principles of complex analysis and its applications. Fourier transform, Laplace transform and Z-transform are treated in complex field. Finally random processes (stacinary, markovian, spectral density) are treated.
Course outlines:
1. | | Complex plane. Functions of compex variables. Elementary functions. |
2. | | Cauchy-Riemann 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. | | Z-transform 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. |
Exercises outline:
1. | | Complex plane. Functions of compex variables. Elementary functions. |
2. | | Cauchy-Riemann 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. | | Z-transform 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. |
Literature:
[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. |
Requirements:
Informace viz
http://math.feld.cvut.cz/0educ/pozad/b3b01kat.htm
Webpage:
http://math.feld.cvut.cz/hamhalte/A3M01MKI.htm
Subject is included into these academic programs:
Page updated 6.12.2019 17:52:32, semester: Z,L/2020-1, L/2018-9, Z,L/2019-20, Send comments about the content to the Administrators of the Academic Programs |
Proposal and Realization: I. Halaška (K336), J. Novák (K336) |