TU Wien:Mathematik 1 UE (diverse)/Übungen WS06/Beispiel 58

Man berechne ohne Taschenrechner alle Werte von ${\sqrt[{4}]{1+i}}$ in der Form $[r,\Phi ]$.

$z_{1}=1+i=[{\sqrt {2}},{\frac {\Pi }{4}}]$

Nützliches

z = $[r,\Phi ]$

${\sqrt[{n}]{z}}=[{\sqrt[{n}]{r}},{\frac {\Phi }{n}}+{\frac {2k\pi }{n}}]$

Lösung

$w_{1}=[{\sqrt[{8}]{2}},{\frac {\Pi }{16}}]$

$w_{2}=[{\sqrt[{8}]{2}},{\frac {\Pi }{16}}+{\frac {\Pi }{2}}]$

$w_{3}=[{\sqrt[{8}]{2}},{\frac {\Pi }{16}}+\Pi ]$

$w_{4}=[{\sqrt[{8}]{2}},{\frac {\Pi }{16}}+{\frac {3\Pi }{2}}]$