These are selected frames from an animation in which the circles in the left figure (the domain) move to the right. The middle figure in each frame is 1/z applied to the domain. The right figure is also 1/z, but drawn on the Riemann sphere.
On the plane, 1/z can appear extraordinarily complicated - watch what happens to the complement of the image of the disk. It seems to become the interior! On the sphere the action is quite simple.(Question: is there any reason to vary the function to some other linear fractional transformation?)
