I have to find image of $\{z\mid Re(z)\leq 0\}$ under a complex bilinear transformation which maps points $0,-i,\infty$ to $10,5-i5,5+i5$ respectively.
I am thinking like, as all points $0,-i,\infty$ are on imaginary axis and that $10,5-i5,5+i5$ on the circle $|z-5|=5$ in clockwise direction . Now as we walk on imaginary axis from $0$ to $-i$, the region $\{z\mid Re(z)\leq 0\}$ is to the right and hence image will be in right hand if we walk clockwork around circle $|z-5|=5$, so required image is $|z-5|<5$. Am I correct ? Please suggest me and I need proper result regarding this . In Book “Visual Complex analysis” by Needham at page number $149$, it is given related to this but I am unable to understand. Thank you.