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

$w_0 = [\sqrt[8]{2}, \frac{\pi}{16}]$

$w_1 = [\sqrt[8]{2}, \frac{\pi}{16} + \frac{\pi}{2}]$

$w_2 = [\sqrt[8]{2}, \frac{\pi}{16} + \pi]$

$w_3 = [\sqrt[8]{2}, \frac{\pi}{16} + \frac{3\pi}{2}]$