Q:

All Values of a Complex expression

I am asked to find all values to $$\left(\frac{1-i}{\sqrt2}\right)^{1+i}$$
I do not know how to approach a power with complex part. Any help would be appreciated.

A:

Since 
$$ (1-i) = \sqrt{2}\,e^{-\pi i/4} $$
we have:
$$ (1-i)^i = e^{\frac{i}{2}\log 2+2k\pi} e^{\pi/4} = (\cos(\log\sqrt{2})+i\sin(\log\sqrt{2}))\,e^{2k\pi+\pi/4}$$
with $k\in\mathbb{Z}$.

