Question

The first time I tried to understand Euler's formula was about 2 years ago. I didn't need it, I just randomly run across it, when trying to understand a Fourier transformation. The problem was, that I did'nt know what expomentiation to 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 exponentioation by complex numbers? If it is a theorem, where can I find something about complex exponents?

Answer 1

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.

Answer 2

I don't think this is really an MO-appropriate question, but since Andreas thought it was worth an answer, I'll follow up with a pointer to this note that I wrote up for an inquisitive student many years ago; note that it's basically an expansion of Andreas's answer.

And as long as I'm citing myself, here is a blog post that might help.

Answer 3

As it was stated above, the answer depends on the choice of exposition. My favorite way is to define $e^z$ by a power series. Then Euler's formula becomes the DEFINITION of sine and cosine. You quickly derive all properties of sine and cosine from it. And then, if the students already know some other definition of sine and cosine, you show that this is the same thing.

Another approach is assuming that definitions and properties of sine and cosine were already established, one can define $e^z$ by the Euler formula, and derive all properties of the exponential from the properties of sine and cosine. This way is usually chosen by the authors of very elementary CV texts. But I like the first way.

In both variants Euler's formula is a definition. But of course, one can define sine and cosine and exponential separately and independently, and then prove Euler's formula as a theorem. (This is what Euler probably did).