Solution
= Solution
The <complex exponential> satisfies $e^{iz}=e^{-y}(\cos x+i\sin x)$. Multiplying by $x+iy$ gives
$$
f(z)=e^{-y}\big[(x\cos x-y\sin x)+i(x\sin x+y\cos x)\big].
$$
Thus its <real part> is
$$
\boxed{\operatorname{Re}f=e^{-y}(x\cos x-y\sin x)}.
$$