Find the value of the given expression:
$$(1+i)^i$$
$$= e^{ln(1+i)^i}$$
$$= e^{i\cdot ln(1+i)}$$
$$= e^{i\cdot (ln|1+i|+i\cdot arg(1+i))}$$
$$= e^{i\cdot (ln\sqrt {2}+i(\frac{\pi}{4}+2k\pi))} (k\in \mathbb{Z})$$
$$= e^{i\cdot (ln\sqrt {2})} \cdot e^{-(\frac{\pi}{4}+2k\pi))}$$
$$=(\cos{ln\sqrt {2}}+i\sin{ln\sqrt {2}})\cdot e^{-(\frac{\pi}{4}+2k\pi))}$$
