Difference between $\frac{1}{z}$ and $\frac{1}{\bar{z}}$ transformation

Question

I read a lot of courses about transformation and sometimes for example to transform a line not passing through the center of inversion to a circle they use $\frac{1}{z}$ or $\frac{1}{\bar{z}}$. My question is which difference does it represent?

Answer 1

We have $ \frac {1}{\overline {z}}= \frac {1}{\overline {z}}\cdot \frac {z}{z}= \frac{z}{\left| z\right| ^{2}}$ and therefore $\left| z\right| \cdot \left| \frac {1}{\overline {z}}\right| =\frac {\left| z\right| ^{2}}{|z|^{2}} =1$. So $\frac {1}{\overline {z}}$ is the inversion of the point $z$ in the unit circle, whereas $\frac {1}{z}$ is the inversion $ \frac {1}{\overline {z}}$ followed by conjugation, that is, reflection with respect to the real axis.