Find the images under the transformation $w=f(z)$

Question

(a) For the transformation $w=z+\bar z$ find the image of $D=\{z\in C| |z|=1$ and $Im(z)\geq 0\}$

(b) For the transformation $w=2iz+i$ find the image of $D=\{z\in C| |z|\lt1$ and $Re(z)\gt 0\}$

For (a) let $z=a+ib$, set $D$ this mean semi-circle as image, click here

let $F$ represented image of $D$ under transformation $w=z+\bar z, \;w=u+iv$

I have

$$u+iv \in F \iff u+iv=z+\bar z\iff u+iv=2a+i0$$

Thus, $-2\leq u\leq 2$ and $v= 0$

Hence image of $D$ As image,click here

For (b) let $z = a+ib$, set $D$ this mean semi-circle as image, click here

let $F$ represented image of $D$ under transformation $w=2iz+i$,$w=u+iv$

I have $\displaystyle w=u+iv \in F \iff |w-i|=|2iz| \iff |w-i|=2|z|\iff \frac{|w-i|}{2}=|z|$

Thus, $\displaystyle \frac{|w-i|}{2} \lt 1 $ then $|w-i| \lt 2$

Hence, $\displaystyle u+iv \in F\iff u^2+(v-1)^2 \lt 4$ and $Im(z)>i$

Therefore image of $D$ as image,click here(overlap between red and blue color)

Is it correct? I'm not sure please help me.

Thank you.

Answer 1

Your both are not correct.

for (a) just observe what is $z+\bar z?$ It is $2\mathcal Re (z)$. So if $|z|=1$ what are the possible values of $\mathcal Re (z)$?

also for (b) observe what does the transformation do. multiplying $iz$ rotate $z$ by an angle of $\frac{\pi}{2}$ counterclockwise around the origin. multiplying by $2$, scale $|iz|$ with a factor of $2$. then adding $i$ will shift $2iz$, $1$ unit upward. so what would be the end result?

By the way, your diagrams for Domains are correct. So try to think geometrically.

Answer 2

Your solution for $(a)$ is not correct. My hint would be to notice that $z+\overline{z} = 2\Re(z)$ for all $z\in\Bbb C$.

The solution for $(b)$ is also not correct. My hint would be to take it in strides.

1. What is the visual/geometrical effect of the transformation $z \mapsto iz$?
2. What is the visual/geometrical effect of the transformation $z \mapsto 2z$?
3. What is the visual/geometrical effect of the transformation $z \mapsto z + i$?

Figure out each of them and apply them to your initial domain $D$ in succession.