## 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.

### 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  .....