How to Evaluate \(i^{i}\)

Conventionally, we knew that any number raised to real power can be evaluated, and its value might be real or complex. For instance, when we say \(i^{2}\) we mean that this value is equivalent to \(i\times i\) which equal to \(-1\).  

But, what does it mean when we say: "Multiply the imaginary number \(i\) \(i-th\) times" ? 

The easy answer is to use the natural logarithm, .i.e. 

    \(\Rightarrow\) \(i^{i} = e^{ln\left ( i \right )^{i}}\)

    \(\Rightarrow\)\(e^{ln\left ( i \right )^{i}} = e^{i ln\left ( i \right )}\)

    \(\Rightarrow\)\( e^{i ln \left ( i \right )} = e^{\left ( i \right )\left ( \frac{\pi}{2} \right )} \)  click here for more details.

Hence,  \( i^{i} = e^{-\frac{\pi }{2}} \)

So, in general, we can ask this key fact question, which is: is it true that any  \(complex^{complex} \in \mathbb{R}\) ? 

To answer this question , we do need to do the following  .....