I'm confused on a question I just come across, I' only recently started working with complex numbers and do not know how to go about answering these questions :
(a) Give in an Argand diagram a geometric description of all complex numbers z such that $ \bar z $= $z^{-1}$
(b) Express the reflection across the line Re(z)=Im(z), the diagonal of the Argand diagram, as an operation with complex numbers, combining multiplication and conjugation. Hint: you know that complex conjugation is just reflection about the real axis in the Argand diagram. Try to find a way to rotate the Re(z)=Im(z) diagonal onto the real axis to perform a reflection, then rotate back.
Hints:
a) $\bar z=z^{-1}$ is equivalent to $z\bar z=1$, i.e. $|z|^2=1\iff |z|=1$.
b) A rotation which moves a point on the diagonal of the Argand plane onto the real axis corresponds to multiplication by $\mathrm e^{-\tfrac{i\pi}4}$. So you have to take the conjugate of $z\mkern1.5mu \mathrm e^{-\frac{i\pi}4}$, then perform the reverse rotation, i.e. multiply by $\mathrm e^{\tfrac{i\pi}4}$.