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QUESTION Express each of the following complex numbers in
Cartesian form:

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\item[(a)]
 $e^{1+\frac{i\pi}{6}}.$

\item[(b)]
 $\textrm{Log}(1+i).$

\item[(c)]
 $i^\frac{i}{\pi}$

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ANSWER
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\item[(a)]
$e^{1+i\frac{\pi}{6}}=e(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}=\frac{e\sqrt{3}}{2}+\frac{ei}{2}.$

\item[(b)]
Log$(1+i)=\log|1+i|+i$Arg$(1+i)=\log \sqrt{2}+i\frac{\pi}{4}.$

\item[(c)]
$i^\frac{i}{\pi}=e^\frac{i}{\pi}\log
i=e^{\frac{i}{\pi}(i\frac{\pi}{2}+2n)},\ (n\in \textbf{Z}),\
(e^{-\frac{1}{2}}$ is the principle value).

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