Is this a homework problem?
The figure has an inconsistency. The left hand side shows a light blue region whose corners are z1, z2, and the origin. Points z3 and z4 are not corners of the light blue region on the left. On the right hand side, z3 and z4 are corners of the light blue region. Are you seeking a map that sends the light blue region on the left onto the light blue region on the right? Or are you seeking a map that sends (z1, z2, z3, z4) to (z1', z2', z3', z4')? I think it would be more clear if the light blue region on the left had the same corners as the light blue region on the right.
There are twio questions implied in your question:
- What is the conformal map that does the desired mapping?
- How can one implement the conformal map of quesiton 1 as a matlab script?
Do you know the answer to question 1?
