# sketch given region an its image under given mapping $2 \leq Im z \leq 5$ and $w=iz$

sketch given region an its image under given mapping

$2 \leq Im z \leq 5$ and $w=iz$

Here is what I got so far

$z=x+iy$ so $2\leq y\leq 5$ and $w=-y+ix$ sp $-5+ix \leq w \leq 2+ix$

this implies that $-5 \leq Re( w) \leq -2$. But the answer is the back of the book says the answer is $-5 \leq Re(z) \leq -2$. I'm not sure I understand how they got this, and I don't understand how they determine the image from this.

I understand that this map send a disk with radius $1$ to another unit disk. I wonder if anyone would please explain this very carefully for me please. I really want to understand conformal mapping stuff.

$$w=iz=i(x+iy)=-y+ix = u+iv$$
$$u=-y, v=x$$
$$2\ge \operatorname{Im} Z\ge5 \Rightarrow 2\ge y\ge5 \Rightarrow -2\le-y\le-5 \Rightarrow-2\le u\le -5 \Rightarrow -5\ge u\ge-2$$
Remember that: $$-x<6$$ becomes $$x>-6$$ or $$-5x<-10$$ becomes $$x>2$$
Therefore, $$'u'$$ moves from $$-5$$ to $$-2$$. Sorry for bad editing...