If we have $\displaystyle w = \frac{1}{z} $, we can think of this as the following: $\displaystyle W = \frac{1}{|z|^{2}} z $ and $\displaystyle Z = \overline{W} $.

But can we also think about it as $\displaystyle W = \frac{1}{|z|^{2}} \overline{z} $ without the second transformation?