CVEL9

From Exampleproblems

$(1+i)^i\,$ on the principal branch.

$(1+i)^i = e^{i\,{\rm Ln}(1+i)}$

${\rm Ln}(1+i) = \ln|1+i| + i\,{\rm Arg}(1+i) = \ln\sqrt{2} + i\,\frac{\pi}{4}$

Therefore,

$(1+i)^i = e^{i\,{\rm Ln}(1+i)} = e^{i\left(\ln(\sqrt{2}) + i\,\pi/4\right)} = e^{-\pi/4}\left(\cos(\ln\sqrt{2}) + i\,\sin(\ln\sqrt{2})\right)$