TU Wien:Algebra und Diskrete Mathematik UE (diverse)/Übungen SS19/Beispiel 36

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

$\phi$ = arctan(b/a)

z = $[r,\phi ]$

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

Lösung

$w_{0}=[{\sqrt[{8}]{2}},{\frac {\pi }{16}}]$

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

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

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