Complex Analysis II (M7270) Course Information
Prerequisits. Linear Algebra I and II, Mathematical Analysis III and IV, Complex
Analysis I.
Lecturer. Doc. Ilya Kossovskiy (kossovskiyi@math.muni.cz)
Course credits: 4
Timetable: Wed, Fri 12:00 – 13:50, M3. Lectures start Fri, Sep 20.
Overview. Complex Analysis is one of the classical and possibly most beautiful areas of
mathematics, which has numerious applications in Geometry, Analysis, Number Theory,
Applied Mathematics and Physics. Complex Analysis studies functions of (one or
several) complex variables satisfying the complex differentiability condition
(holomorphic functions). It turns out that the latter simple condition forces unbelievably
strong properties of a holomorphic function, such as its infinite differentiability and
analyticity (local representation as a sum of a power series). The special feature of
Complex Analysis is that very strong results, e.g. the Riemann Mapping Theorem
("every proper simply connected domain in complex plane is conformally equivalent to
the unit disc"), appear in an extremely elegant way from a sequence of simple
geometric/analytic statements, each of which can be exposed to the students "in one
line". This illuminates the effectiveness of Complex Analysis in application to other
fields, where Complex Analysis gives a simple geometric picture for numerous difficult
and important phenomena.
The course is dedicated to studying in detail advanced aspects of Complex Analysis
which fall out of the Complex Analysis I course. At a glance, it is structured as follows.
We first study advanced topics from Complex Analysis in one variable: the theory of
conformal mappings, Schwarz reflection principle, the argument principle and its
applications, multi-valued analytic functions and Riemann surfaces, Riemann mapping
theorem, Runge approximation theorem, harmonic functions of two variables.
Next, we study selected topics from Complex Analysis in Several Variables:
holomorphic functions of several variables, Hartogs theorem, analytic continuation,
envelopes of holomorphy, holomorphic mappings, basics of the d-bar theory.
Syllabus:
- Brief overview of the Complex Analysis I course
- Riemann sphere. Linear-fractional mappings.
- Conformal mappings of domains. The Schwarz reflection principle.
- Schwarz Lemma. Automorphisms of model domains in the Riemann sphere.
- Analytic continuation. Multi-valued analytic functions. Isolated singularities. Riemann
surfaces.
- The argument principle, Rouche's Theorem, and applications. Openness principle,
Hurwitz's theorem.
- Families of holomorphic functions. Compactness principle. Riemman Mapping
Theorem.
- Additional topics: the modular function, Small Picard Theorem, Runge Theorem,
Weierstrass Factorization Theorem.
- Harmonic functions in two variables. Dirichle Problem.
- The geometry of complex space. Holomorphic functions in several variables. Cauchy
formula. Power series expansion. Hartogs separate analyticity theorem.
- The phenomenon of forced analytic continuation. Holomorphic convexity. Analytic
discs. Envelopes of holomorphy.
- Holomorphic mappings. Schwarz Lemma. Automorphism groups of domains in
complex space.
- Basics of the d-bar theory.
Literature.
[1] Gamelin, Theodore W. Complex analysis. (English) Undergraduate Texts in
Mathematics. Springer-Verlag, New York, 2001.
[2] Shabat, B. V. Introduction to complex analysis. Part II. Functions of several
variables. (English) Translated from the third (1985) Russian edition by J. S. Joel.
Translations of Mathematical Monographs, 110. American Mathematical Society,
Providence, RI, 1992.
[3] Chabat, B. Introduction à l'analyse complexe. Tome 1. (French) Fonctions d'une
variable. Translated from the Russian by Djilali Embarek. Traduit du Russe:
Mathématiques. "Mir'', Moscow, 1990.