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QUESTION

Write $-1$ in complex exponential form and hence, or otherwise,
find all values of $\ln(-1)$.

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ANSWER


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For $z=-1, r=1,\theta=\pi+2n\pi$

So $-1=1e^{j(\pi+2n\pi)},\ n=0,\pm1,\pm2,\ldots$

Therefore
$\ln(-1)=\ln\{1e^{j(\pi+2n\pi)}\}=\ln1+\ln\{e^{j(\pi+2n\pi)}\}$

i.e. $\ln(-1)=j(\pi+2n\pi),\ n$ any integer.



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