\(c\) 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.