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I have the following statements written down in my notes. But I can't see what is happening to get from the second last line to the last line.

$$e^{\operatorname{Log}(z)} = e^{\log(|z|) + i\operatorname{Arg}(z)} = e^{\log(|z|)}[\cos(\operatorname{Arg}(z)) + i\sin(\operatorname{Arg}(z)] = z$$

Anyone know how the second last line is able to reduce into just $z$ on the final line?

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it's from the initial $e^{\log(z)}$ I think – Raymond Manzoni Apr 18 '12 at 0:03
What is $Re[z]$ and $Im[z]$ in terms of $|z|$ and the angle $z$ makes with the positive real axis? It's just polar coordinates, basically. – Mr. F Apr 18 '12 at 0:03

suppose $z=re^{i\theta}$,then $e^{log(|z|)}=r$,$[Cos(Arg(z)) + iSin(Arg(z)]=e^{i\theta}$,

so $e^{log(|z|)}[Cos(Arg(z)) + iSin(Arg(z)]=z$.

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