How do I compute the image of the following function (Linear fractional transformation)

Question

I have the following problem:

We have given the linear fractional transformation $$f(z)=\frac{z+1}{z-1}$$ and the area $A=\{|z|\leq 1, \mathfrak{Im}(z)\geq 0\}$. I need to compute $f(A)$.

So I know that $A$ contains all points of the upper half of the complex unit circle including the real line from $-1$ to $1$ and also the boundary $|z|=1$. Then I thought that maybe it would be easier to write those points in the following form $$z=re^{i\Theta}$$ where $r\in [0,1]$ and $\Theta \in [0,\pi]$. But somehow I don't see what to do if I plug this into $f$ instead of $z$.

Could maybe someone help me here?

Thanks for your help

Answer 1

HINT...Let $$w=\frac{z+1}{z-1}$$ and rearrange to get $$z=\frac{w+1}{w-1}$$

Now applying the condition $$|z|\leq1\implies |w+1|\leq|w-1|$$ to obtain a region in the image plane.

Also by writing $w=u+iv$, you can apply the condition $IM(z)\geq0$ to get another condition on $u$ and $v$.