Find \(\Re\qty{z+2\over z-1}\).
Solution
\[ {z+2\over z-1}\cdot{\bar{z-1} \over \bar{z-1}} &= {(z+2)\bar{z-1} \over \abs{z-1}^2} \\ &= {((z-1) + 3)\bar{z-1} \over \abs{z-1}^2} \\ &= {\abs{z-1}^2 + 3\bar{z} -3 \over \abs{z-1}^2} \\ &= {(x-1)^2 + y^2 + 3(x-iy) - 3 \over (x-1)^2 + y^2} \\ &= {(x-1)^2 + 3x - 3 + y^2\over (x-1)^2 + y^2} + i{-3y\over (x-1)^2 + y^2} \\ &= {(x-1)(x+2) + y^2\over (x-1)^2 + y^2} + i{-3y\over (x-1)^2 + y^2} .\]