The course is over, thanks for participation! The results of the tests a=
re found in the register. The maximum amount of extra points from the

e=
xercises was 10 (from the two tests 24+24). The maximum amount of extra poi=
nts from the exercises for a final exam ('loppukoe') is 6.

SUGGESTIONS FOR SOLUTIONS:

=20Function theory I

=2010 op.

=20Advanced studies.

=20Several books are mentioned at the lectures. However, nearly basic book =
on function theory (of which there are plenty!)

cover most of the lectu=
res.

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.

=20Two 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.

=20Did you forget to register? Mit=C3=A4 tehd=C3=A4.

=20
=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 |

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

=20Wednesday 20.1: Local invertibility of analytic maps at points of confor=
mality. Local mapping properties.

Analytic maps are open maps. Removabl=
e singularities.

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.

=20Tuesday 9.2: The residue theorem. Applications to integrals of rational = functions.

=20Wednesday 10.2: The residue thm (cont.). Computation of trigonometric in= tegrals and other examples. The argument principle.

=20Tuesday 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.

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.

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.

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.

Tuesday 2.3: Normal families. Montel's theorem.

=20Wednesday 3.3: Conformal equivalence of domains. Conformal bijections of=
the unit disc onto itself. Simply connected domains revisited.

The Rie=
mann mapping theorem.

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

=20Wednesday 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.

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

=20Wednesday 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.

=20Tuesday 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.

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

=20Tuesday 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.

=20Tuesday 27.4 Subharmonic functions. Maximum principles. Characterization=
via a maximum principle. Sign of the Laplacian of

a subharmonic functi=
on.

Wednesday 28.4 (last lecture) Subharmonic functions (continued). Propert=
ies of Perron families.

Discussion of possible further topics in comple=
x analysis.

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.