M6170 Complex Analysis

Faculty of Science
Spring 2006
Extent and Intensity
4/2/0. 6 credit(s) (fasci plus compl plus > 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
Contact Person: doc. RNDr. Josef Kalas, CSc.
Timetable
Wed 8:00–9:50 UM, Fri 8:00–9:50 UM
  • Timetable of Seminar Groups:
M6170/01: Fri 10:00–11:50 UM, J. Kalas
Prerequisites
( M3100 Mathematical Analysis III || M4502 Mathematical Analysis 3 ) && 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 course is concentrated particularly to integration in C and Cauchy's theory, properties of holomorphic functions, calculus of residues and its applications, entire and meromorphic functions and elements of conformal mapping theory.
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. 6. Entire functions - definition and classification of entire functions, infinite products of complex numbers and functions, Weierstrass' theorems, order of an entire function, Hadamard's theorem. 7. Meromorphic functions - Logarithmic derivative, argument principle, Rouché's theorem. Meromorphic function in a region, uniqueness theorem. Meromorphic function in C, Mittag-Leffler theorems, Cauchy's theorem on the expansion of meromorphic functions. 8. Introduction to the conformal mapping theory - fractional linear map, conformal mapping and its properties, the main problem of conformal mapping, Riemann mapping theorem, principle of one-to-one correspondence of boundaries, reflection principle, Schwarz-Christoffel theorem.
Literature
  • 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
Assessment methods (in Czech)
Výuka: přednáška, klasické cvičení. Zkouška: písemná a ústní.
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 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 2024, Spring 2025.
  • Enrolment Statistics (Spring 2006, recent)
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