F5066 Functions of complex variable

Faculty of Science
Spring 2020
Extent and Intensity
2/2/0. 4 credit(s). Type of Completion: z (credit).
Teacher(s)
prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Bc. Tomáš Řiháček, Ph.D. (seminar tutor)
Guaranteed by
prof. RNDr. Jana Musilová, CSc.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.
Supplier department: Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Prerequisites
Fundamentals of analysis of real variables
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The discipline is a part of the fundamental course of mathematical analysis for students of physics. The main goals are as follows:

* To present to students fundamentals of the theory of functions of a complex variable and show them specific principial differencies between this theory and the theory of functions of real variables.
* To show students a practical use of the theory, especially the residue theorem and Laplace transformation, for calculating complex as well as real integrals, and mainly for physical applications (quantum physics, solid state physics).

Absolving the course students obtain following knowledge and skills:
* Understanding of fundemantals of theory of functions of a complex variable and differencies of this theory compared with the theory of functions of real variables.
* Practical skills in calculus of functions of a complex variable, especially in calculating complex as well as real integrals using the Cauchy theorem and the residue theorem.
* Practical skills in the use of calculus of functions of a complex variable in physical applications (residue theorem, Laplace transformation).
* Understanding of response functions in physics through the Laplace transformation.
Learning outcomes
Student obtains fundamental theoretical knowledge in this specific theme of mathematical analysis, practical skills in solving mathematical tasks as well as the ability to apply them in solving physical problems. He/she is well prepared for the study of advanced topics in this area of mathematical analysis and mathematical physics.
Syllabus
  • 1.Introductory concepts -- functions of a complex variable, integral.
  • 2. Holomorphic functions, Cauchy-Riemann conditions.
  • 3. Regulární funkce, Taylorova řada.
  • 4. Cauchy theorem and its use for calculating integrals.
  • 5. Uniqueness theorem, holomorphic prolongation.
  • 6. Applications of uniqueness theorem, elementary functions defined by series.
  • 7. Physical applications of Cauchy theorem and uniqueness theorem, Kramers-Kronig relations.
  • 8. Laurent series and residues.
  • 9. Theorem of residues and it consequences.
  • 10. Applications of residues theorem for calculating integrals.
  • 11. Multivalued function, prolongations along curves, fundamental multivalued functions.
  • 12. Laplace transformation, response functions.
  • 13. Applications of Laplace transformation in physics.
  • 14. Conformal mapping and its applications.
Literature
  • JEVGRAFOV, Marat Andrejevič. Funkce komplexní proměnné. Translated by Ladislav Průcha. Vyd. 1. Praha: SNTL - Nakladatelství technické literatury, 1981, 379 s. URL info
  • JEVGRAFOV, Marat Andrejevič. Sbírka úloh z teorie funkcí komplexní proměnné. Translated by Anna Něničková - Věra Maňasová - Eva Nováková. Vyd. 1. Praha: SNTL - Nakladatelství technické literatury, 1976, 542 s. URL info
  • RUDIN, Walter. Analýza v reálném a komplexním oboru. Vyd. 2., přeprac. Praha: Academia, 2003, 460 s. ISBN 8020011250. info
Teaching methods
Lectures: theoretical explanation with practical examples
Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems
Assessment methods
Teaching: lectures, consultative exercises,
Exam - credit: written test (two parts: (a) solving problems, (b) test).
Current requirements: Written tests. The presence in exercises is obligatory (75 %).
Language of instruction
Czech
Further comments (probably available only in Czech)
The course can also be completed outside the examination period.
The course is taught once in two years.
The course is taught: every week.
General note: S.
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 1999, Autumn 2000, Autumn 2001, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Spring 2012, Autumn 2011 - acreditation, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, spring 2018, Spring 2021, Spring 2023, Spring 2025.
  • Enrolment Statistics (Spring 2020, recent)
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