Evaluate $(1+i)^{-i}$

Question

$(1+i)^{-i}$

I need to find all the possible values

$r=\sqrt{1^2+1^2}=\sqrt{2}$

$\theta=\frac{\pi}{4}$

$[\sqrt{2}e^{\frac{\pi}{4}i}]^{-i}$

How to continue from here?

Answer 1

\begin{align}\bigl(\sqrt2e^{\frac{\pi i}4}\bigr)^{-i}&=\exp\left(-i\left(\log\bigl(\sqrt2\bigr)+\frac{\pi i}4+2k\pi i\right)\right)\\&=\exp\left(-\frac{\log2}2i+\frac\pi4+2k\pi\right)\\&=e^{\frac\pi4+2k\pi}\left(\cos\left(\frac{\log2}2\right)-i\sin\left(\frac{\log2}2\right)\right),\end{align}with $k\in\mathbb Z$. If you just want to use the principal value of the logarithm, take $k=0$.