Given a non-zero complex number $z=x+i y$, where $x$ and $y$ are real, find expressions for the real and imaginary parts of the following functions of $z$ in terms of $x$ and $y$ :

(i) $e^{z}$,

(ii) $\sin z$

(iii) $\frac{1}{z}-\frac{1}{\bar{z}}$,

(iv) $z^{3}-z^{2} \bar{z}-z \bar{z}^{2}+\bar{z}^{3}$,

where $\bar{z}$ is the complex conjugate of $z$.

Now assume $x>0$ and find expressions for the real and imaginary parts of all solutions to

(v) $w=\log z$.