The first time I tried to understand [Euler's formula](http://en.wikipedia.org/wiki/Euler%2527s_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?



Is Euler's formula a theorem or a definition?
The question seems to presuppose that there is only one way to develop this part of complex analysis. There are several, among which some may take Euler's formula as a definition of complex exponentiation. As far as I know, most define complex exponentiation in a different way --- by the Taylor series of the exponential function (as in Rudin's "Real and Complex Analysis") or by a differential equation. In such a development, Euler's formula might still be a definition, namely a definition of the cosine and sine, or, if the trigonometric functions have already been defined another way, then Euler's formulas would be a theorem. 


