Technology

Ausgewählte Kapitel aus der Funktionentheorie by Wolfgang Fischer, Ingo Lieb

By Wolfgang Fischer, Ingo Lieb

Inhalt
Vorwort - Leitfaden - Hermitische Metriken und normale Familien - Analytische Fortsetzung und Riemannsche Flächen - Harmonische Funktionen und das Dirichlet-Problem - Der Uniformisierungssatz - Funktionentheorie im Einheitskreis - Spiegelungsprinzip und Dreiecksfunktionen - Hilberträume und konforme Abbildungen - Literaturverzeichnis

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Extra info for Ausgewählte Kapitel aus der Funktionentheorie

Sample text

Es sei G offener Tei! einer Riemannschen Flache x. Eine Funktion f: G ....

Inf {lwl: wE r - {On und beliebiges z ist P IDe (z) bijektiv. SchlieBlich ist X zusammenhangend (als stetiges BUd von cr) und Hausdorffsch: Sind Xl = Zl + r und X2 = Z2 + r verschiedene Punkte in X, so haben die Mengen Zl und Zz + r einen positiven Abstand 25 in cr, und pDo (zd, pDo (Z2) sind disjunkte Umgebungen von Xl und X2. Wir definieren nun eine komplexe Struktur auf X: Flir jedes offene V C cr, das dUTCh p injektiv abgebildet wird, ist (p I V)-l: p V ~ V eine Karte. ir je zwei soIehe Karten (p I Vlr l und (p I V2 )-1 ist (p I Vlr l 0 (p I Vz ) auf jeder Wegkomponente von (p I VZ)-l (p VI n P V2 ) eine Translation aus r, also biholomorph.

2. i) Es sei r: [a, b] ~ ce ein Weg von Zo nach z lund f E 19zo . Dann gibt es hOchstens einen Weg 1: [a, b] ~ 19(: mit 1(a) = fund po 1 =r. ii) Es sei weiter f holomorph auf De (zo) und llings r analytisch fortsetzbar zu g. Dann gibt e~ einen in Pzo (f) beginnenden Weg in 19er mit p 0 =r. Der Endpunkt von r ist PZI (g). iii) Es sei ein Weg in 19er von f E 19zo nach g E 19Z 1' f: De (zo) ~ ce und g: D'r/ (ZI) ~ ce seien holomorphe Reprlisentanten von fund g. durch analytische Fortsetzung llings des Weges r = p 0 1.

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