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 .....

26/08/2017 05:14 297