$c$ Goes to Heaven

Looking at,

$c^k= e^{-i\cfrac{\pi}{2k}}. c^{i\cfrac{\pi}{2}}$ This is wrong.

Consider,

$y=c^{i\cfrac{\pi}{2}}$

$ln(y)=i\cfrac{\pi}{2}ln(c)$

$y=c.e^{i\cfrac{\pi}{2}}$

$c^k= c.e^{i\cfrac{\pi}{2}\left(1-\cfrac{1}{k}\right)}$

Visually,

$\cfrac{\pi}{2}\left(1-\cfrac{1}{k}\right) \longrightarrow \cfrac{\pi}{2}$   as $k\rightarrow \infty$

With $|c|\rightarrow \infty$,  $c$ goes to heaven, upwards.