M3100F Mathematical Analysis III

Faculty of Science
autumn 2021
Extent and Intensity
4/2/0. 6 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. Mgr. Petr Zemánek, Ph.D. (lecturer)
Mgr. Pavla Musilová, Ph.D. (seminar tutor)
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
Wed 12:00–13:50 M1,01017, Fri 12:00–13:50 M1,01017
  • Timetable of Seminar Groups:
M3100F/01: Thu 9:00–10:50 F4,03017, P. Musilová
M3100F/02: Thu 11:00–12:50 F4,03017, P. Musilová
Prerequisites
( M2100F Mathematical Analysis II || M2100 Mathematical Analysis II ) && ! M3100 Mathematical Analysis III
The knowledge from courses Mathematical Analysis I and II is assumed.
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
This is the final part of the three-semester course of mathematical analysis. It is focused on the topics concerning infinite series, analysis in the complex domain, and functional analysis. After passing the course, the student will be able:
define and interpret the notions from the theory of infinite series and complex and functional analysis;
formulate relevant mathematical theorems and to explain methods of their proofs;
analyze problems from the topics of the course;
understand to theoretical and practical methods of the theory of infinite series and complex and functional analysis;
apply the methods of mathematical analysis to concrete problems.
Learning outcomes
At the end of the course students will be able to:
define and interpret the notions from the theory of infinite series and complex and functional analysis;
formulate relevant mathematical theorems and to explain methods of their proofs;
analyze problems from the topics of the course;
understand to theoretical and practical methods of the theory of infinite series and complex and functional analysis;
apply the methods of mathematical analysis to concrete problems.
Syllabus
  • I. Infinite series: foundations, series with nonnegative terms, alternating series, series with arbitrary terms, criteria of convergence, absolute and relative convergence, operations with infinite series, Riemann rearrangement theorem, product of infinite series.
  • II. Sequences and infinite series of functions: pointwise and uniform convergence, criteria and examples of uniform convergence, differentiation and integration of sequence or series of functions, power series and their applications, radius of convergence, Taylor series, Fourier series, Fourier transformation.
  • III. Introduction to complex analysis: foundations, sequences and infinite series in C, differentiation in C, Cauchy-Riemann equations, holomorphic functions, functions given by power series, exponential functions, trigonometric functions, complex power and logarithmic functions, integration in C, Cauchy theorem and its applications to integrals in real domain, Cauchy integral formula, properties of holomorphic functions, Laurent series, isolated singularities, residue of function at point, residue theorem, computations of integrals by using residue theorem.
  • IV. Foundations of functional analysis: Hilbert space, linear functionals, dual spaces, self-adjoint linear operators in Hilbert spaces, spectral theory.
Literature
  • KALAS, Josef. Analýza v komplexním oboru. 1. vyd. Brno: Masarykova univerzita, 2006, iv, 202. ISBN 8021040459. info
  • DOŠLÁ, Zuzana and Vítězslav NOVÁK. Nekonečné řady. Vyd. 1. Brno: Masarykova univerzita, 1998, 113 s. ISBN 8021019492. info
  • KOLMOGOROV, Andrej Nikolajevič and Sergej Vasil‘jevič FOMIN. Základy teorie funkcí a funkcionální analýzy. Translated by Vladimír Doležal - Zdeněk Tichý. Vyd. 1. Praha: SNTL - Nakladatelství technické literatury, 1975, 581 s. info
Teaching methods
Standard theoretical lectures with excercises.
Assessment methods
Lectures: 4 hours/week. Seminars: 2 hours/week.
3 written tests in seminars during the semester (evaluated by 0-5 points; at least 1 point in the case when you reach not less than 30 percents of the overall evaluations; points 1-5 are distributed according to the key 10-15-35-25-15).
Final exam (only if you have at least 1 point from the seminar): Written-test (with numerical examples, max. 25 points) and, only if you reach at least 13 points at all from the seminar and written-test, oral exam (two topics are discussed).
To pass: at least 13 points at all from the seminar and written-test and not to show a noticeable ignorance of basic notions and statements. If you have only 13-16 points, the knowledge in the oral part has to be above-average.
Results of the tests written during the semester are included in the overall evaluation.

The conditions (especially regarding the form of the tests and exam) will be specified according to the epidemiological situation and valid restrictions.
Language of instruction
Czech
Further Comments
Study Materials
The course is taught annually.
Listed among pre-requisites of other courses
The course is also listed under the following terms Autumn 2019, Autumn 2020, Autumn 2022, Autumn 2023, Autumn 2024.
  • Enrolment Statistics (autumn 2021, recent)
  • Permalink: https://is.muni.cz/course/sci/autumn2021/M3100F