Limit Point and Closure

\( ln\left ( i \right ) = i\frac{\pi}{2} \)

To prove the upper identity, we aim to use the Euler's Formula for the complex numbers

 \(e^{i \theta} = cos(\theta) + isin(\theta)\). Since,  \( ln\left ( i \right ) = i\frac{\pi}{2} \) can be interpreted to \(e^{something} = i\) and this will prompt us to choose value of \(\theta\) that serve our need from Euler's Formula. So, by setting the value of \(\theta = \frac{\pi}{2}\), the right hand side of Euler's formula becomes:

            \(e^{i \frac{\pi}{2}} = cos(\frac{\pi}{2}) + isin(\frac{\pi}{2})\)

            \(\Rightarrow \) \(e^{i \frac{\pi}{2}} = 0 + i\left ( 1 \right )\)

Hence , \( ln\left ( i \right ) = i\frac{\pi}{2} \).