PřF:M6170 Complex Analysis - Course Information
M6170 Complex Analysis
Faculty of ScienceSpring 2025
- Extent and Intensity
- 2/2/0. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
In-person direct teaching - 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 - 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
- Mathematics (programme PřF, N-MA)
- Mathematics (programme PřF, B-MA)
- Statistics and Data Analysis (programme PřF, B-MA)
- 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)
- The course is taught annually.
The course is taught: every week. - 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.
- Enrolment Statistics (recent)
- Permalink: https://is.muni.cz/course/sci/spring2025/M6170