Complex Analysis II by Berenstein C.A. (ed)

By Berenstein C.A. (ed)

Lecture notes in arithmetic No.1276

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Weiter bezeichne rr den Kreis vom Radius r um a, orientiert im positiven Sinn (s. Fig. 3c). h. r Jr f(z) dz = z-a i, i g(z) dz, f(z) dz. ,z-a (1) 44 5. Komplexe Integration Indem wir im rechtsstehenden Integral den Integrationsweg r r parametrisieren, rr: t~ z(t):=a+reir, 0

Also ist der grösste Kreis um 0, in dessen Innerem f analytisch ist, der Einheitskreis (s. Fig. 5b).

3f enthaltenden Gebiet analytisch. Ir z - a für jeden im Innern von r liegenden Punkt a den Wert f(a) hat. Welches ist der Wert des Integrals, wenn a im Äusseren von r liegt? 4. Anwendungen der Cauchyschen Integralformel Mittelwerteigenschaft Die Funktion f sei analytisch in einem einfach zusammenhängenden Gebiet G, a sei ein Punkt in G und r ein in G enthaltener, positiv orientierter Kreis um a vom Radius r (s. Fig. 4a). Nach der Cauchyschen Integralformel gilt f(a)=~ 2m Wir parametrisieren f fez) dz.

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