Toronto Math Forum
MAT3342018F => MAT334Tests => Quiz2 => Topic started by: Victor Ivrii on October 05, 2018, 06:13:30 PM

Find the values(s) of the given expression:
\begin{equation*}
\exp \Bigl[\pi \Bigl(\frac{i+1}{\sqrt{2}}\Bigr)^4\Bigr].
\end{equation*}

$(\frac{i+1}{\sqrt{2}})$ in polar coordinates is $\cos(\frac{\pi}{4}) + i\sin(\frac{\pi}{4})$
The roots of this equation are equally spaced on the unit circle around the origin, and the polar angle of $(\frac{i+1}{\sqrt{2}})^4$ is $\pi$
Therefore we know that $(\frac{i+1}{\sqrt{2}})^4 = \cos(\pi) + i\sin(\pi) = 1$
So finally, $\exp[\pi(\frac{i+1}{\sqrt{2}})^4] = \exp[\pi]$