SUBJECT

Chapters of complex function theory

practical

master

Faculty of Science

Mathematics MSc

6

Semesters 1-4

Autumn/Spring semester

The aim of the course is to give an introduction to various chapters of functions of a complex variable. Some of these will be further elaborated on, depending on the interest of the participants, in lectures, seminars and practices to be announced in the second semester. In general, six of the following, essentially self-contained topics can be discussed, each taking about a month, 2 hours a week.

Topics:

• Phragmén-Lindelöf type theorems.
• Capacity. Tchebycheff constant. Transfinite diameter. Green function. Capacity and Hausdorff measure. Conformal radius.
• Area principle. Koebe’s distortion theorems. Estimation of the coefficients of univalent functions.
• Area-length principle. Extremal length. Modulus of  quadruples and rings. Quasiconformal maps. Extension to the boundary. Quasisymmetric functions. Quasiconformal curves.
• Divergence and rotation free flows in the plane. Complex potencial. Flows around fixed bodies.
• Laplace integral. Inversion formuli. Applications to Tauberian theorems, quasianalytic functions, Müntz’s theorem.
• Poisson integral of L_p functions. Hardy spaces. Marcell Riesz’s theorem. Interpolation between L_p spaces. Theorem of the Riesz brothers.
• Meromorphic functions in the plane. The two main theorems of the Nevanlinna theory.

• M. Tsuji: Potential Theory in Modern Function Theory, Maruzen Co., Tokyo, 1959.
• L.V. Ahlfors: Conformal Invariants, McGraw-Hill, New York, 1973.
• Ch. Pommerenke: Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.
• L.V. Ahlfors: Lectures on Quasiconformal Mappings, D. Van Nostrand Co., Princeton, 1966. W.K.Hayman: MeromorphicFunctions, Clarendon Press, Oxford 1964.
• P. Koosis: Introduction to H_p Spaces, University Press, Cambridge 1980.
• G. Polya and G. Latta: Complex Variables, John Wiley & Sons, New York, 1974.