Author Topic: LEC0101 - Quiz 2 D  (Read 335 times)

Kuba Wernerowski

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LEC0101 - Quiz 2 D
« on: October 02, 2020, 11:16:10 AM »
$\textbf{Problem}$ (3pt). Find all the value(s) of the given expression $$ i^{\sqrt{3}}.$$
\begin{align*}\
    i^{\sqrt{3}} &= e^{\ln(i)^{\sqrt{3}}} \\
    &= e^{\sqrt{3} \ln(i)} \\
    &= e^{\sqrt{3}(\ln \lvert{i}\rvert + i \arg{i})} \qquad \quad \ln(i) = \ln \lvert i \rvert + i \arg{i} \text{ since } i \in \mathbb{C}.\\
    &= e^{\sqrt{3}(\ln{1} \, + \, i \left(\frac{\pi}{2} + 2\pi k \right))} \qquad \, k \in \mathbb{Z} \\
    &= e^{\sqrt{3}(i \left(\frac{\pi}{2} + 2\pi k \right))} \\
    &= \cos{\sqrt{3} \left(\frac{\pi}{2} + 2 \pi k \right)} + i \sin{\sqrt{3} \left(\frac{\pi}{2} + 2 \pi k \right)}
\end{align*}
« Last Edit: October 02, 2020, 11:22:04 AM by Kuba Wernerowski »