What is the image of $\{z:Im(z)>0\} $ under $f(z)=\frac{1+z}{1-z}$?
Attempt:
I have evaluated $f$ at three points $0,-1,$ and $1$.
We have $f(0)=1$, $f(-1)=0$ and $f(1)=\infty$. To me it seems the image should be the circle of radius $1$. But I feel like it is wrong since $i$ is in the plane $Im(z)>0$ and $f(i)=i$ not inside the circle of radius one. What I am I missing conceptually? How do I determine the image ? Thanks!
Best Answer
This map is a continuous bijection of the Riemann sphere with itself.
The image of the (extended) real line is the (extended) real line.
So the image of the upper half plane is the component of the complement of the real line containing $f(i)$.
So it is either the upper or the lower half plane. (Which is it?)
You will have to adjust the domain and/or range depending on whether you want to include $\infty$.