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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.

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$$w=iz=i(x+iy)=-y+ix = u+iv$$

$$u=-y, v=x$$

$$2\ge \operatorname{Im} Z\ge5 \Rightarrow 2\ge y\ge5 \Rightarrow -2\le-y\le-5 \Rightarrow-2\le u\le -5 \Rightarrow -5\ge u\ge-2$$

Remember that: $-x<6$ becomes $x>-6$ or $-5x<-10$ becomes $x>2$

Therefore, $'u'$ moves from $-5$ to $-2$. Sorry for bad editing...

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  • $\begingroup$ Welcome to MSE. Please use MathJax to format both, your answers and your questions :) $\endgroup$ – mrtaurho Oct 10 '18 at 13:49

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