# Subject description - AE3M01MKI

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AE3M01MKI Mathematics for Cybernetics
Roles:P, V Extent of teaching:4P+2S
Department:13101 Language of teaching:EN
Guarantors:  Completion:Z,ZK
Lecturers:  Credits:8
Tutors:  Semester:Z

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:

  S.Lang. Complex Analysis, Springer, 1993.  L.Debnath: Integral Transforms and Their Applications, 1995, CRC Press, Inc.  Joel L. Shiff: The Laplace Transform, Theory and Applications, 1999, Springer Verlag.

Requirements: