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Integral von Potenzen des Cosinus

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Exercise

https://texercises.com/exercise/integral-von-potenzen-des-cosinus/

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Exercise:
Zeige _^pi cost^n rm dt fracpi^n binomnn durch Auswertung des Integrals _|z| frac z leftz+fraczright^n rm dz.

Solution:
renewcommandimathrmi renewcommanddmathrmd renewcommandemathrme Mit der Parametrisierung des Einheitskreises ze^i t für tinpi erhalten wir _|z| frac z leftz+fraczright^n rm dz _^pi e^-i tlefte^i t + e^-i tright^n i e^i t d t i ^n _^pi cost^n d t : i ^n I. Ausserdem gilt frac z leftz+fraczright^n fracz^+^nz^n+. Die Cauchy-Integralformel CIF lautet f^kz frack!pii _partial D fracfomegaomega-z^k+. In unserem Fall ist fomega omega^+^n limits_k^n binomnk omega^n-k qquad mn qquad z. Die n-te Ableitung ist f^nomega limits_k^n binomnk fracn-k!n-k! omega^n-k. An der Stelle omega verschwinden alle Terme ausser der mit Potenz was gerade für kn der Fall ist. Deswegen gilt f^n n!binomnn. Eingesetzt in die CIF ergibt sich n!binomnn fracn!piii^nI. Umformen führt zu I fracpi^n binomnn.

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Integrale mit Funktionentheorie by pw

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Tags	binomialkoeffizient, cauchy, cosinus, formel, integral, kreisintegral
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Difficulty	(4, default)
Points	4 (default)
Language	(Deutsch)
Type	Calculative / Quantity
Creator	pw
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