MAT334-2018F > Quiz-2

Q2 TUT 0202

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

Quentin King:
$(\frac{i+1}{\sqrt{2}})$ 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, $\exp[\pi(\frac{i+1}{\sqrt{2}})^4] = \exp[-\pi]$