xⁱ? It's mind boggling and interesting. It's a little more complicated than your normal exponent, but i can be saved in this predicament by another constant; e.
     e is useful here because e^x can be written as:

     x⁰/0! + x¹/1! + x²/2! + x³/3! + x⁴/4! + x⁵/5! + x⁶/6! + ...

     Therefore eⁱ is:

     i⁰/0! + i¹/1! + i²/2! + i³/3! + i⁴/4! + i⁵/5! + i⁶/6! + ...

     or

     1/0! + i/1! - 1/2! - i/3! + 1/4! + i/5! - 1/6! + ...

     Using this for eⁱ, we can find xⁱ by using the following equation:

     xⁱ = (e^{ln x})ⁱ = (eⁱ)^{ln x} = e^{i*(ln x)}

     We can even generalize this for x^y where y is any complex number.

     x^y = (e^{ln x})^y = (e^y)^{ln x} = e^{y*(ln x)}

     or

     (y*ln x)⁰/0! + (y*ln x)¹/1! + (y*ln x)²/2! + (y*ln x)³/3! + (y*ln x)⁴/4! + (y*ln x)⁵/5! + (y*ln x)⁶/6! + ...

     If you know how to find the natural logarithm of a complex number, then you can say x is a complex number too.

     Now, what does it all mean?

That's odd?

Calculating ln(i)

What's It Good For?

xi	Calculating the Trig Functions

More Resources

ii Explained By Dr. Math