By William A. Veech

Author William A. Veech, the Edgar Odell Lovett Professor of arithmetic at Rice college, offers the Riemann mapping theorem as a different case of an lifestyles theorem for common masking surfaces. His concentrate on the geometry of advanced mappings makes widespread use of Schwarz's lemma. He constructs the common overlaying floor of an arbitrary planar zone and employs the modular functionality to boost the theorems of Landau, Schottky, Montel, and Picard as results of the life of sure coverings. Concluding chapters discover Hadamard product theorem and major quantity theorem.

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**Extra info for A second course in complex analysis**

**Sample text**

K fOr n E lN o. Sie ist genau dann gleichmaBig konvergent in X. falls die Foige (L ~ = 0 fn )k der Partial- summen eine Cauchyfolge ist. Das bedeutet: Zu e > 0 gibt es ein Ne E IN. Das bedeutet: Zu e > 0 gibt es ein Ne E IN. 4 so dass fOr aile k 2: m 2: Ne gilt: Existenz konvergenter Teilfolgen Die folgenden beiden Siitze liefern Kriterien fOr die Existenz konvergenter Teilfolgen von Folgen komplexer Zahlen (Satz von Bolzano-WeierstraB) bzw. Foigen holomorpher Funktionen (Satz von Montel). 5 II' II bezeichnet die Supremumsnorm auf X.

3. 5. 4 Satz: Unendlich haufige reelle Differenzierbarkeit Eine harmonische Funktion)st beliebig oft reell differenzierbar. 5 Satz: Maximum- und Minimumprinzip Maximum- und Minimumprinzip fOr beliebige Gebiete Sei G ein Gebiet in CC. und die harmonische Funktion u: G -+ IR nehme in G ein lokales Maximum oder Minimum an. Dann ist u konstant. le6 Maximum- und Minimumprinzip fOr beschrankte Gebiete Sei G ein beschranktes Gebiet in CC und die stetige Funktion u: G -+ IR sei auf G harmonisch. 6 fOr aile z E G.

ZEK Wegen der 2-Periodizitat ist Mauch das Maximum von S ,= {z EO::: 11m zl :5 1}. ) aus o::\S gilt die Abschatzung (vgl. 3 iv»: IQ