M6170 Complex Analysis

Faculty of Science
Spring 2015
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. RNDr. Josef Kalas, CSc. (lecturer)
Guaranteed by
doc. RNDr. Josef Kalas, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 8:00–9:50 M2,01021
  • Timetable of Seminar Groups:
M6170/01: Mon 10:00–11:50 M2,01021, J. Kalas
Prerequisites
( M3100 Mathematical Analysis III || M4502 Mathematical Analysis 4 ) && 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.
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. Complex functions - continuous functions, complex differentiability, Cauchy-Riemann equations, holomorphic functions. Series of functions, power series. Elementary functions, power, n-th root, exponential, logarithmic, goniometric, cyclometric, hyperbolic, hyperbolometric functions, generalized power. 3. Integral, Cauchy theory - curves in C, complex integration, primitive, path dependence. Cauchy's theorem, Cauchy integral formulas. 4. Properties of holomorphic functions - Liouville's theorem, Cauchy's inequality, Morera's 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
  • KALAS, Josef. Analýza v komplexním oboru (Complex Analysis). 1st ed. Brno: Masarykova univerzita v Brně, 2006, 202 pp. ISBN 80-210-4045-9. info
  • ČERNÝ, Ilja. Analýza v komplexním oboru. 1. vyd. Praha: Academia, 1983, 822 s. info
  • 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
  • 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
Teaching methods
lectures and class exercises
Assessment methods
Examination: written and oral. One written test will be realized during the semester. It is required to obtain at least half of the total amount of points. The exam is composed of a written and an oral part. The written part consists of four exercises. It is necessary to obtain at least 1,5 from possible 4 points.
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught annually.
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 2016, Spring 2017, spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2024, Spring 2025.
  • Enrolment Statistics (Spring 2015, recent)
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