1. ## Complex transformation query

Let the region R be defined by $\displaystyle |z|<1$, if z is any point in the region R, and $\displaystyle z^*$ is its conjugate, find the corresponding region for w where

$\displaystyle w=zz^*$

My solution was this. I know that $\displaystyle zz^*=|z|^2$ and since the modulus of a complex number is a real number its square will be also. So we know w is a real number and so will lie on the real line. In addition its modulus will be less than one, so I think w lies on the real axis between 0 and 1.

However the book says that it's any point inside the unit circle centered at the origin (i.e. no change). But wouldn't this be true for $\displaystyle |w|=zz^*$?? The book definitely asks for $\displaystyle w=zz^*$, so is there a typo on the book or have I made a mistake?

Regards,

Stonehambey

2. Originally Posted by Stonehambey
Let the region R be defined by $\displaystyle |z|<1$, if z is any point in the region R, and $\displaystyle z^*$ is its conjugate, find the corresponding region for w where $\displaystyle w=zz^*$

However the book says that it's any point inside the unit circle centered at the origin (i.e. no change). But wouldn't this be true for $\displaystyle |w|=zz^*$?? The book definitely asks for $\displaystyle w=zz^*$, so is there a typo on the book or have I made a mistake?
I agree with you. The way the problem defines $\displaystyle w$, we get $\displaystyle w\in \Re~\&~0\le w<0$.