Function theory II, spring 2010

What is going on?

We just decided that Exam 1 of the course takes place on Tuesday 23.3 at 14-17 in C323, a DIFFERENT ROOM from Tuesday's lectures. The last thing included is the
Riemann mapping theorem. Note that there are no lectures during the week of the Exam 1.

Exercises

Exercises 1 (29.01)
Exercises 2 (12.02)
Exercises 3 (19.02)
Exercises 4 (26.02)
Exercises 5 (05.03)
Exercises 6 (19.03)
Exercises 7 (02.04)

SUGGESTIONS FOR SOLUTIONS:

Solutions 1
Solutions 2
Solutions 3
Solutions 4
Solutions 5

Lecture notes (scetches only)

Lectures (-17.03)

Prerequisities

Function theory I

Scope

10 op.

Type

Advanced studies.

Literature

Several books are mentioned at the lectures. However, nearly basic book on function theory (of which there are plenty!)
cover most of the lectures.

Lecturer

Eero Saksman

Lectures

At least the weeks 3-4, 6-9, 11, 13, 15-17 tu 14-16 C124, we 10-13 C123, in addition exercise groups 2 hours weekly.
First lecture 19.01.

Eastern holidays 1.-7.4.

Test

Two exams. The first test takes place on 23.3 at 14-17 in C323, not the same room where we have Tuesdays lectures. The last thing included is the
Riemann mapping theorem.
The time of the second test will be decided later on.

Register

Did you forget to register? Mitä tehdä.

Exercise groups

Ryhmä	Päivä	Aika	Paikka	Pitäjä
1.	Fri	10-12	B322	Jarmo Jääskeläinen

Logbook

Tuesday 19.1: general things, recalling FT 1. Analyticity of uniform limits.

Wednesday 20.1: Local invertibility of analytic maps at points of conformality. Local mapping properties.
Analytic maps are open maps. Removable singularities.

Tuesday 26.1: Removable singularities (cont.). Poles. Essential singularities. Weierstrass theorem on behaviour close to an essential singularity.

Wednesday 27.1: Analytic continuations. Laurent series. Residue at a singularity.

Tuesday 9.2: The residue theorem. Applications to integrals of rational functions.

Wednesday 10.2: The residue thm (cont.). Computation of trigonometric integrals and other examples. The argument principle.

Tuesday 16.2: Rouche's theorem. New proof of openess of analytic functions and the fundamental theorem of algebra. Injectivity of
limits of sequences of injective maps.

Wednesday 17.2: The gamma function: functional equation and the meromorphic extension to complex plane.
Gauss product formula and other basic formulas. Nonvanishing of the Gamma function. The Riemann zeta function.
Euler product formula for the zeta function.

Tuesday 23.2: Non-vanishing of zeta function for Re(s)>1. Hankel integral formula and analytic (meromorphic) continuation of the Riemann
zeta function tothe whole complex plane.

Wednesday 24.2: The Riemann functional equation. Discussion of Riemann hypothesis and prime number theorem.
Proof of (one direction) of the exact relation between the Riemann hypothesis and prime number theorem.

Tuesday 2.3: Normal families. Montel's theorem.

Wednesday 3.3: Conformal equivalence of domains. Conformal bijections of the unit disc onto itself. Simply connected domains revisited.
The Riemann mapping theorem.

Tuesday 16.3: Comments on Riemann's mapping theorem. Homotopy of paths and its basic properties.

Wednesday 17.3: Independence of the fundamental group on
the base point. Invariance in homeomorphisms of the domain. Any homotopy can be expressed as a combination on finitely many elementary transformations.
Integral of an analytic function over general continuous curve.