I want to show this mapping from the upper half space to the unit ball is a bijection:
So one thing I tried is to just use the direct definitions of being injective and surjective, but it didn't take too long to get an algebraic mess. But then I was starting to suspect this mapping is in fact the real and imaginary parts of some Mobius transform in complex analysis. However, finding out the coefficients of the Mobius transform is also not an easy task, but is there a way to do so?
Best Answer
A nice shortcut for this is to just set $y=0$, i.e. restricting $f$ to $\mathbb{R}$ when you think of it as a function $\mathbb{C}\to\mathbb{C}$. This tells you that $$f(x)=\frac{2x}{x^2+1}+\frac{x^2-1}{x^2+1}i=i\frac{x^2-2ix-1}{x^2+1}=i\frac{(x-i)^2}{x^2+1}=i\frac{x-i}{x+i}$$ for $x\in\mathbb{R}$. So, if $f$ is indeed a Möbius transformation, it must be $f(z)=i\frac{z-i}{z+i}$. Now you can just do the algebra to check that this really is true.