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!

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. 
