# Trying to figure out a complex equality

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.