10 de septiembre

De Moivre para Potencias
𝑧 = 𝑟 𝑛 (cos(𝑛𝜃 ) + 𝑖 sin(𝑛𝜃 ))
𝑛
De Moivre para Raíces
1
1
𝜃 + 2𝜋𝑘
𝜃 + 2𝜋𝑘
) + 𝑖 sin (
))
𝑤 = 𝑧 𝑛 = 𝑟 𝑛 (cos (
𝑛
𝑛
𝟏
𝟑
√𝟒 − 𝟒√𝟑𝒊 = (𝟒 − 𝟒√𝟑𝒊)𝟑 ;
𝒏 = 𝟑, 𝒌 = 𝟎, 𝟏, 𝟐
2
𝑟 = √(4)2 + (−4√3) = √16 + 16(3) = √64 = 8
−4√3
𝜋
) = tan−1 (−√3) = −60° → −
4
3
8(cos(60) − 𝑖 sin(60))
𝜋
𝜋
8 (cos ( ) − 𝑖 sin ( ))
3
3
1
60 + 360𝑘
60 + 360𝑘
) − 𝑖 sin (
))
83 (cos (
3
3
𝜋
𝜋
+
2𝜋𝑘
+ 2𝜋𝑘
1
83 (cos ( 3
) − 𝑖 sin ( 3
))
3
3
𝜃 = tan−1 (
𝑘=0
+ 360(0)
60 + 360(0)
) − 𝑖 sin (
))
3
3
60
60
𝑤 = 2 (cos ( ) − 𝑖 sin ( ))
3
3
𝑤 = 2(cos(20) − 𝑖 sin(20))
𝑤 = 1.879 − 0.684𝑖
𝑤=
1
60
83 (cos (
𝑤=
1
60
83 (cos (
𝑤=
1
60
83 (cos (
𝑘=1
+ 360(1)
60 + 360(1)
) − 𝑖 sin (
))
3
3
420
420
) − 𝑖 sin (
))
𝑤 = 2 (cos (
3
3
𝑤 = 2(cos(140) − 𝑖 sin(140))
𝑤 = −1.532 − 1.285𝑖
𝑘=2
+ 360(2)
60 + 360(2)
) − 𝑖 sin (
))
3
3
780
780
) − 𝑖 sin (
))
𝑤 = 2 (cos (
3
3
𝑤 = 2(cos(260) − 𝑖 sin(260))
𝑤 = −0.347 + 1.969𝑖