Find a linear fractional transformation that maps $Re(z) = 1/2$ onto the circle $|w-4i| = 4$
My first attempt tries to map this through simple intuitive transformation. I am starting with a vertical line crossing $z = 1/2$ transforming this to the center $T(z) = z - 1/2$ I can now inverse it to get a circle:
$$T(z) = \frac{2}{2z-1}$$
Since the circle has radius 2, and is two units upwards I can then get:
$$T(z) = \frac{4}{2z-1} + 4i$$
However, this is not equal to the suggested answer of $w = 4i - 4 + 4/z$. Does this type of intuition work for simply mappings? Could someone point out the flaw in logic?
The transformation $w=\frac1z$ transforms the line $Re(z)=\frac12$ into the circle centred at $1$ with radius $1$.
You may wish to check this for yourself.
Follow this with a translation, so that the transformation $w=\frac1z-1$ will transform the line $Re(z)=\frac12$ into the circle centred at the origin with radius $1$.
Fiollow this with an enlargement of factor $4$, so that the transformation $w=4(\frac1z-1)$ will transform the line $Re(z)=\frac12$ into the circle centred at the origin with radius $4$.
Finally do another translation so the centre of the circle is moved to $4i$.
Therefore the complete transformation is $$w=4(\frac1z-1)+4i=4i-4+\frac4z$$