The first time I tried to understand Euler's formula was about 2 years ago. I didn't need it, I just randomly ran across it, when trying to understand a Fourier transformation. The problem was, that I didn't know what exponentiation by complex number means.

Now I have some free time and I would like to understand it again. So the first question, is it a theorem of some theory, or is it a definition of exponentiation by complex numbers? If it is a theorem, where can I find something about complex exponents?