complex numbers - What is the meaning of Euler's identity?

I know that euler's identity state that $e^{ix} = \cos x + i\sin x$

But e is a real number. What does it even mean to raise a real number to an imaginary power. I mean multiplying it with itself underoot $-1$ times? What does that mean?

Answer

If $z$ and $w$ are complex numbers, you define $z^w = e^{w \log z}$. The problem is that $\log w$ assumes several values, so you can say that $z^w$ is a set. So if you fix a principal value for ${\rm Log}\,z$, you have a principal power $e^{w\,{\rm Log}\,z}$. For each branch you'll have a different power.

More exactly, the argument of a complex number is the set: $$\arg z = \{ \theta \in \Bbb R \mid z = |z|(\cos \theta + i \sin \theta) \}.$$We call ${\rm Arg}\,z$ the only $\theta \in \arg z$ such that $-\pi < \theta \leq \pi$. Also, if $z \neq 0$, we have: $$\log z = \{ \ln |z| + i \theta \mid \theta \in \arg z \}.$$
Call ${\rm Log}\,z = \ln |z| + i \,{\rm Arg}\,z$. Then you could say that $z^w = \{ e^{w \ell} \mid \ell \in \log z \}$.

To make sense of $e^{\rm something}$, we use the definition of the exponential with series.