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Functions of Complex Variable and Integral Transformations

* Exchange students do not have to consider this information when selecting suitable courses for an exchange stay.

Course Unit Code470-4109/01
Number of ECTS Credits Allocated6 ECTS credits
Type of Course Unit *Compulsory
Level of Course Unit *Second Cycle
Year of Study *First Year
Semester when the Course Unit is deliveredWinter Semester
Mode of DeliveryFace-to-face
Language of InstructionCzech
Prerequisites and Co-Requisites There are no prerequisites or co-requisites for this course unit
Name of Lecturer(s)Personal IDName
SIM46Mgr. Lenka Přibylová, Ph.D.
HOR33doc. Ing. David Horák, Ph.D.
LAM05prof. RNDr. Marek Lampart, Ph.D.
KAL0063prof. RNDr. René Kalus, Ph.D.
Summary
Functions of complex variable and integral transformations are one of the basic tools of effective solution of technical problems. The students will get knowledge of basic concepts
of functions of complex variable and integral transformations.
Learning Outcomes of the Course Unit
To give students knowledge of basic concepts of complex functions of complex variable and integral transformations.
Course Contents
Lectures:
Complex functions and mappings. Complex differentiation, contour integration and deforming the contour.
Complex series: power series, Taylor and Laurent series. Residue theorem. Applications.
Introduction to Fourier series. Orthogonal systems of functions. Generalized Fourier series. Applications.
Introduction to integral transforms. Convolution.
Laplace transform, fundamental properties. Inverse Laplace transform. Applications.

Fourier transform, fundamental properties. Inverse Fourier transform. Applications.
Z-transform, fundamental properties. Inverse Z-transform. Applications.


Exercises:
Practising of complex functions, linear and quadratic mappings.
Practising of complex differentiation, conformal mappings, contour integration and deforming the contour.
Examples of Taylor and Laurent series and applications.
Examples of orthogonal systems of functions, Fourier series and applications.
Practising of Laplace transform. Solution of differential equation.
Practising of Fourier transform and examples.
Practising of Z-transform. Solution of difference equation.


Projects:
Two individual works and their presentation on the theme:

Fourier series.
Laplace transform.
Recommended or Required Reading
Required Reading:
Needham, Tristan, Visual Complex Analysis, Oxford University Press, 2023, ISBN: 0192868926
Stewart Ian, Complex Analysis, Cambridge, 2018, ISBN: 9781108436793
Shah, Nita H. and K. Naik, Monika. Integral Transforms and Applications, Berlin, Boston: De Gruyter, 2022. https://doi.org/10.1515/9783110792850
Kozubek, T., Lampart, M.: Integral Transforms, 2022, https://homel.vsb.cz/~lam05/Teaching.html
Bouchala, J., Lampart, M.: An Introduction to Complex Analysis, 2022, https://homel.vsb.cz/~lam05/Teaching.html
Needham, Tristan, Visual Complex Analysis, Oxford University Press, 2023, ISBN: 0192868926
Stewart Ian, Complex Analysis, Cambridge, 2018, ISBN: 9781108436793
Shah, Nita H. and K. Naik, Monika. Integral Transforms and Applications, Berlin, Boston: De Gruyter, 2022. https://doi.org/10.1515/9783110792850
Kozubek, T., Lampart, M.: Integrální transformace, 2012, http://mi21.vsb.cz/modul/integralni-transformace
Bouchala, J.: Funkce komplexní proměnné, 2012, http://mi21.vsb.cz/modul/funkce-komplexni-promenne
Recommended Reading:
Howie J.M., Complex Analysis, Springer-Verlag London, 2003, ISBN 978-1-85233-733-9.
Abdon Atangana, Ali Akgul, Integral Transforms and Engineering, Taylor & Francis Ltd, 2023, ISBN-13 9781032416830
Howie J.M., Complex Analysis, Springer-Verlag London, 2003, ISBN 978-1-85233-733-9.
Abdon Atangana, Ali Akgul, Integral Transforms and Engineering, Taylor & Francis Ltd, 2023, ISBN-13 9781032416830
Planned learning activities and teaching methods
Lectures, Tutorials, Project work
Assesment methods and criteria
Task TitleTask TypeMaximum Number of Points
(Act. for Subtasks)
Minimum Number of Points for Task Passing
Exercises evaluation and ExaminationCredit and Examination100 (100)51
        Exercises evaluationCredit40 (40)20
                Sem. pr. 1Project10 0
                Sem. pr. 2Project10 0
                Test 1Written test10 0
                Test 2Written test10 0
        ExaminationExamination60 11