Why is Euler's formula a definition?

Question

Even though there are proofs for Euler's formula for complex exponentials (see wikipedia for instance), it is mentioned as a "definition" in most textbooks. Why is that? My understanding is that a definition defines something that cannot be derived from other theorems.

Answer 1

The complex exponential function needs some definition.

There are several ways one could define it:

1. Power series
2. As a solution to a differential equation
3. As the unique holomorphic extension of the real exponential function to the whole plane
4. Via the limit definition $\lim_{n \to \infty} (1+\frac{z}{n})^n$
5. Via Euler's formula
6. ...

One can take any one of these as the definition. Then you should probably prove that the other properties hold. What is a definition and what is a theorem is, in a sense, a matter of taste.

Answer 2

I believe there might be mixing-up of some ideas. It's not the formula per se the one that needs a definition for the formula. It is the expression $e^x$. This expression, as mentioned by Steven, may be defined in various ways, which all represent the same thing. It is like the expression $e^x$ is some kind of ubiquitous expression, if you may see it that way. You may enunciate it in various ways, which in the end mean the same thing.

In the case of Euler's Formula, what we find is yet another (yet incredible) way to describe the idea of $e^x$. The formula may have been derived, but once again, it comes down to Earth to present another way of defining $e^x$!