Since z = a^2/z on the circle, we see that w as given by (1) is
purely
real on the circle C and therefore if* = 0. Thus C is a
streamline.
If the point z is outside 0, the point az
/z is inside 0, and vice-versa. Since
all the singularities off(z) are by hypothesis exterior to C,
all the singularities
off(a?/z) are interior to C ; in particular f(a
z
/z) has no singularity at infinity,
since f(z) has none at z = 0. Thus w has exactly the same
singularities as/(z)
and so all the conditions are satisfied.