Value of sine of complex numbers I stumbled upon a problem with evaluating the sine function for complex arguments.
I know that in general I can use 
$$
\sin(ix)=\frac{1}{2i}(\exp(-x)-\exp(x))=i\sinh(x).
$$
But I could also write the sine function as the imaginary part of the exponential function as 
$$\sin(ix)=\text{Im}(\exp(i(ix)))=\text{Im}(\exp(-x))=0$$
where Im is the imaginary part.
Well, apparently I am not allowed to write it like that, but I don't see why. Could you give me a hint what went wrong here?

First, note that
$$\mbox{for }y\in\mathbb{R}\ \mbox{we have}\ \sin{y}=\mathrm{Im}\{e^{iy}\}.$$
Stating $\mathrm{Im}\left\{\mathrm{exp}(-x)\right\}=0$ means you assume $x\in\mathbb{R}$. However, then $ix$ is complex, such that you cannot apply the above rule.
