M6170 Complex Analysis

Faculty of Science
Spring 2024
Extent and Intensity
2/2/0. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
Teacher(s)
doc. Mgr. Petr Zemánek, Ph.D. (lecturer)
Guaranteed by
doc. Mgr. Petr Zemánek, Ph.D.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 19. 2. to Sun 26. 5. Wed 12:00–13:50 M4,01024
  • Timetable of Seminar Groups:
M6170/01: Mon 19. 2. to Sun 26. 5. Thu 12:00–13:50 M3,01023, P. Zemánek
Prerequisites
( M3100 Mathematical Analysis III || M4502 Mathematical Analysis 4 || M3100F Mathematical Analysis III ) && M2110 Linear Algebra II
Mathematical analysis: Differential calculus of functions of one and several variables, integral calculus, sequences and series of numbers and functions, metric spaces. Linear algebra: Systems of linear equations, determinants, matrices, linear spaces, linear transformation.
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
Complex analysis is a classic part of mathematical analysis. It has various smart and often unexpected applications in many fields of mathematics. It is an effective tool even outside the mathematics, especially in physics and engineering. The basic goal of the course is to familiarize the students with the fundamentals of the theory of functions of one complex variable, especially with integration in C and Cauchy's theory, properties of holomorphic functions, calculus of residues and its applications.
Learning outcomes
After passing the course, the student will be able:
to define and interpret the basic notions used in the basic parts of Complex analysis and to explain their mutual context;
to formulate relevant mathematical theorems and statements and to explain methods of their proofs;
to use effective techniques utilized in basic fields of Complex analysis;
to compare differences between the theory of functions of a complex variable and the theory of functions of real variables;
to apply acquired pieces of knowledge for the solution of specific problems including problems of applicative character.
Syllabus
  • 1. Introduction to the discipline: complex numbers, straight line, circle, generalized circle, afinity in C and its special cases, topological concepts, stereographic projection, Gauss and extended Gauss plane, sequences and series of complex numbers.
  • 2. Foundations of complex calculus: continuous functions, complex differentiability, Cauchy-Riemann conditions, holomorphic functions, series of functions and power series, elementary functions (polynomials, exponential, n-th root, logarithm, goniometric, cyclometric, hyperbolic, hyperbolometric functions, generalized power).
  • 3. Integral & Cauchy theory: curves in C, complex integration, primitive function, path dependence, Cauchy theorem, Cauchy integral formulas.
  • 4. Properties of holomorphic functions: Liouville theorem, Cauchy inequality, Morera theorem, sequences and series of holomorphic functions, Taylor expansion, uniqueness theorem, maximum modulus principle.
  • 5. Calculus of residues: Laurent series, isolated singularities, residues, residue theorem, application of the calculus of residues.
Literature
    required literature
  • KALAS, Josef. Analýza v komplexním oboru (Complex Analysis). 1st ed. Brno: Masarykova univerzita v Brně, 2006, 202 pp. ISBN 80-210-4045-9. info
    recommended literature
  • ČERNÝ, Ilja. Analýza v komplexním oboru. 1. vyd. Praha: Academia, 1983, 822 s. 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
    not specified
  • NOVÁK, Vítězslav. Analýza v komplexním oboru. 1. vyd. Praha: Státní pedagogické nakladatelství, 1984, 103 s. info
  • VESELÝ, Jiří. Komplexní analýza. 1st ed. Praha: Univerzita Karlova v Praze, Nakladatelství Karolinum, 2000, 244 pp. ISBN 80-246-0202-4. info
  • LANG, Serge. Complex Analysis. 3rd ed. Springer-Verlag, 1993, 458 pp. ISBN 0-387-97886-0. info
  • JEVGRAFOV, Marat Andrejevič. Funkce komplexní proměnné. Translated by Ladislav Průcha. Vyd. 1. Praha: SNTL - Nakladatelství technické literatury, 1981, 379 s. URL info
  • NAHIN, Paul J. An imaginary tale :the story of [odmocnina z minus jedné]. Princeton, New Jersey: Princeton University Press, 1998, xvi, 257 s. ISBN 0-691-02795-1. info
Teaching methods
lectures and class exercises
Assessment methods
The exam has a written and oral parts.
Language of instruction
Czech
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
Teacher's information
The lessons are usually in Czech or in English as needed, and the relevant terminology is always given with English equivalents. The target skills of the study include the ability to use the English language passively and actively in their own expertise and also in potential areas of application of mathematics. Assessment in all cases may be in Czech and English, at the student's choice.
The course is also listed under the following terms Spring 2008 - for the purpose of the accreditation, Spring 2011 - only for the accreditation, Spring 2002, Spring 2003, Spring 2004, Spring 2005, Spring 2006, Spring 2007, Spring 2008, Spring 2009, Spring 2010, Spring 2011, Spring 2012, spring 2012 - acreditation, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2025.
  • Enrolment Statistics (Spring 2024, recent)
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