What is the mapping that rotates the entire complex plane through an angle $\theta$ about a given point $z_{0}$.
My approach : The mapping $w_{1}=e^{i \theta} z $ rotates the plane through the angle $\theta$ about the origin. In particular, the point $z_{0}$ is mapped to the point $e^{i \theta} z_{0}$.
I'm stuck here on how to establish the mapping to the given question. Any help in solving this is much appreciated.
The mapping $z \mapsto z - z_0$ translates the complex plane so that the point $z_0$ moves to the origin.
Consider applying this translation, then a rotation, then an inverse translation to move everything back in place.