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By Rafinesque C.S

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Ist ζ eine n-te Einheitswurzel, so ist 0 = ζ n − 1 = (ζ − 1)(1 + ζ + ζ 2 + · · · + ζ n−1 ), utzlich, also ζ = 1 oder 1 + ζ + ζ 2 + · · · + ζ n−1 = 0. Diese Gleichung ist sehr n¨ genauso wie die folgende Beziehung: Es ist ζ −1 = 1/ζ = ζ n /ζ = ζ n−1 . Andererseits ist ζ −1 = ζ. Im Falle ζ = cos(2π/n)+ i sin(2π/n) folgt dann: ζ + ζ −1 = 2 cos(2π/n). 5. Beispiele A. Im Falle n = 3 erf¨ ullt eine Einheitswurzel die Gleichung 1 + ζ + ζ 2 = 0. Also ist √ −1 ± i 3 , ζ= 2 √ und deshalb cos(120◦ ) = −1/2 und sin(120◦ ) = 3/2.

Auf ∂D1 (0). Damit muss G = D1 (0) oder = C \ D1 (0) ∪ {− i } sein. Weil T ( i ) = 0 ist, ist T (H) = D1 (0). Als letzte geometrische Anwendung behandeln wir die Berechnung von Einheitswurzeln. Ist ζ eine n-te Einheitswurzel, so ist 0 = ζ n − 1 = (ζ − 1)(1 + ζ + ζ 2 + · · · + ζ n−1 ), utzlich, also ζ = 1 oder 1 + ζ + ζ 2 + · · · + ζ n−1 = 0. Diese Gleichung ist sehr n¨ genauso wie die folgende Beziehung: Es ist ζ −1 = 1/ζ = ζ n /ζ = ζ n−1 . Andererseits ist ζ −1 = ζ. Im Falle ζ = cos(2π/n)+ i sin(2π/n) folgt dann: ζ + ζ −1 = 2 cos(2π/n).

Beispiele A. Im Falle n = 3 erf¨ ullt eine Einheitswurzel die Gleichung 1 + ζ + ζ 2 = 0. Also ist √ −1 ± i 3 , ζ= 2 √ und deshalb cos(120◦ ) = −1/2 und sin(120◦ ) = 3/2.

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