Points $\displaystyle P$ and $\displaystyle Q$ represent the complex numbers $\displaystyle w$ and $\displaystyle z$ respectively in the Argand diagram.

If $\displaystyle w=\frac{1+zi}{z+i}$, $\displaystyle z \neq -i$ and $\displaystyle w=u+iv$, $\displaystyle z=x+iy$, express $\displaystyle u$ and $\displaystyle v$ in terms of $\displaystyle x$ and $\displaystyle y$.

Prove that when $\displaystyle P$ describes the portion of the imaginary axis between the points representing $\displaystyle -i$ and $\displaystyle i$, $\displaystyle Q$ describes the whole of the positive imaginary axis.