Date: Sat, 20 Aug 2022 01:33:57 +0300 (EEST) Message-ID: <1397308024.51855.1660948437604@wiki-1.it.helsinki.fi> Subject: Exported From Confluence MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_Part_51854_815712920.1660948437604" ------=_Part_51854_815712920.1660948437604 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Content-Location: file:///C:/exported.html Function theory II, spring 2010

# = Function theory II, spring 2010

=20

### Exercises

=20

SUGGESTIONS FOR SOLUTIONS:

=20

=20

### Prerequisities

= =20

Function theory I

=20

=20

10 op.

=20

=20

=20

### Literature

=20

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

=20

=20

Eero Saksman

=20

### Lectures

=20

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.

= =20

Eastern holidays 1.-7.4.

=20

### Test

=20

Two exams. Second test is on Thursday 20.5 10-13 in room C123. The solut= ion of Dirichlet's problem in the general domain is not included in the are= a of the test.

=20

### Register

=20

Did you forget to register? Mit=C3=A4 tehd=C3=A4.

=20

### Exercise groups

= =20
=20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 = =20 =20

Ryhm=C3=A4

P=C3=A4iv=C3=A4

Aika

Paikka

Pit=C3=A4j=C3=A4

1.

Fri

10-12

B322

Jarmo J=C3=A4=C3=A4skel=C3=A4inen

=20

### Logbook

=20

Tuesday 19.1: general things, recalling FT 1. Analyticity of uniform lim= its.

=20

Wednesday 20.1: Local invertibility of analytic maps at points of confor= mality. Local mapping properties.
Analytic maps are open maps. Removabl= e singularities.

=20

Tuesday 26.1: Removable singularities (cont.). Poles. Essential singular= ities. Weierstrass theorem on behaviour close to an essential singularity.<= /p>=20

Wednesday 27.1: Analytic continuations. Laurent series. Residue at a sin= gularity.

=20

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

=20

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

=20

Tuesday 16.2: Rouche's theorem. New proof of openess of analytic functio= ns and the fundamental theorem of algebra. Injectivity of
limits of seq= uences of injective maps.

=20

Wednesday 17.2: The gamma function: functional equation and the meromorp= hic extension to complex plane.
Gauss product formula and other basic f= ormulas. Nonvanishing of the Gamma function. The Riemann zeta function.
= Euler product formula for the zeta function.

=20

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

=20

Wednesday 24.2: The Riemann functional equation. Discussion of Riemann h= ypothesis and prime number theorem.
Proof of (one direction) of the exa= ct relation between the Riemann hypothesis and prime number theorem.

=20

Tuesday 2.3: Normal families. Montel's theorem.

=20

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

=20

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

=20

Wednesday 17.3: Independence of the fundamental group on
the base po= int. Invariance in homeomorphisms of the domain. Any homotopy can be expres= sed as a combination on finitely many elementary transformations.
Integ= ral of an analytic function over general continuous curve.

=20

Tuesday 16.3: The homotopic version of Cauchy's theorem. Free homotopy. = Characterizations of simply connected domains.

=20

Wednesday 17.3: Characterizations of simply connected domains (continued= ). E.g., a domain is simply connected iff its boundary is connected. The fu= ndamental group of the punctured plane.

=20

Tuesday 13.4: Definition of harmonic functions. Laplace operator. Connec= tion to analytic functions. Existence of the conjugate function. Regularity= .
Invariance under analytic change of variables.

=20

Wednesday 14.4: Mean value theorem. (local) Sub meanvalue property. Loca= l and strong maximun principles. Uniqueness via boundary values. The Poisso= n formula.

=20

Tuesday 20.4: Fundamental properties of the poisson kernel. Solution of = the Dirichlet problem for the disc. Characterization of harmonicity via the= mean value
principle. Removable singularities. Harnack's inequality.=20

Wednesday 21.4: Harnack's principle. Reflection principles for harmonic = and analytic functions. Conformal maps between annuli.

=20

Tuesday 27.4 Subharmonic functions. Maximum principles. Characterization= via a maximum principle. Sign of the Laplacian of
a subharmonic functi= on.

=20

Wednesday 28.4 (last lecture) Subharmonic functions (continued). Propert= ies of Perron families.
Discussion of possible further topics in comple= x analysis.

=20

Wednesday 12.05 (two extra lectures) Perron's method for solving the Dir= ichlet's problem. Barriers. Solvability for simply
connected domains. S= olvability for domains whose each boundary component consists of more than = one point. Definition and
existence of the Green function.

------=_Part_51854_815712920.1660948437604--