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An answer to a comlex equation I was working on was $z = \frac{1}{2} + \frac{i}{2}$ My teacher further developed it to be $e^{\frac{i\pi}{4}-\frac{1}{2}\ln{2}}$ And here's what I tried: $z = \frac{1}{2} + \frac{i}{2} = z = \frac{1}{\sqrt{2}}e^{\frac{i\pi}{4}} = e^{\frac{1}{2}\ln{2}}e^{\frac{i\pi}{4}} = e^{\frac{1}{2}\ln{2}+\frac{i\pi}{4}}$

I feel this is stupid, but I can't see why we have different answers. Anyone? Thanks!

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The mistake occurs here: $\frac{1}{\sqrt{2}}e^{\frac{i\pi}{4}} = e^{\frac{1}{2}\ln{2}}e^{\frac{i\pi}{4}}.$ In fact, we have $e^{\frac{1}{2}\ln{2}}=2^{\frac{1}{2}}=\sqrt{2}.$ Therefore, we should have $\frac{1}{\sqrt{2}}=(\sqrt{2})^{-1} = e^{-\frac{1}{2}\ln{2}}.$ Mixing this, your answer matches with your teacher's answer.