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?
 A: 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.
A: When you first learn exponential expressions like $b^n$, the exponent $n$ is a natural number and $b^n$ means the product of $b$ with iteself $n$ times.
However, you soon learn that there are more general exponential expressions which do not mean the product of $b$ with itself some number of times. For example, $b^{-3} = 1/b^3$ and $b^0 = 1$ and $b^{2/3} = \sqrt[3]{b^2}$. These formulas with rational number exponents are derived by requiring that basic arithmetic formulas of exponentiation can be extended, namely the formulas $b^x b^y = b^{x+y}$ and $(b^x)^y = b^{xy}$.
Exponential expressions with real-valued exponents, $b^x$, are a further extension so that $b^x$ is a continuous function of $x$, without changing the values of $b^x$ that we already know and love, when $x$ is a positive or negative integer or rational number. Applying calculus we then derive a new formula:
$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots
$$
Exponential expressions with complex-valued exponents, $b^z$, are a further extension so that $b^z$ is a complex-analytic function of $z$, without changing the values we already know when $z=x+0i$ is a real number. Applying complex analysis we then derive a new formula:
$$e^{i\theta} = \cos(\theta) + i \sin(\theta)
$$
To summarize, it's not your mothers's exponentiation. Well, okay, it is your mother's, but maybe it's not your great-great-great-...-great-grandmother's exponentiation. Starting from the naive idea of multiplying a number by itself, by deeply studying the mathematical properties of exponentiation, and by applying tools of calculus and complex analysis, we obtain much more general and sophisticated ideas of exponentiation.
A: I think $e^{a+ib}=e^a(\cos b + i \sin b)$ is "what it means" in the sense that this is the best way to define complex powers to be consistent with their real counterparts. It is consistent with the series expansion of $e^x$ too, so everything joins up.
Of course this raises the issue that $a+bi$ gives the same result as $a+bi +2\pi i$ so the complex logarithm and complex powers have to be carefully handled - there are issues which simply don't arise if you stick with the real numbers. And that leads to considering Riemann Surfaces, and beyond ...
