On my book, it says the sufficient and necessary condition for a Mobius transformation $T:w = \frac{az+b}{cz+d}$ to map the upper half plane $\{z:\mathrm{Im}(z) > 0\}$ to itself is that:
a. $a,b,c,d$ are all real numbers
b. $ad – bc > 0$
I can see the two conditions are sufficient. But I feel confused for the necessity. On my book it says, since $T$ maps real line to real line, thus $a,b,c,d$ are all real numbers. But if $a,b,c,d$ satisfy the condition, how about $ia,ib,ic,id$? They are not real numbers any more. Or, even if it says $T$ could be reduced to the form where they are all real, how to see this? Thank you!
Edit: Actually my confuse is, since my book claims four arguments are real only through one sentence, so I assume there should be a very simple argument that could immediately get to the result that $a,b,c,d$ are real. This is where I'm confused. Thank you!
Best Answer
Choose three different real numbers $z_1,z_2$ and $z_3$, so $T(z_1), T(z_2)$ and $T(z_3)$ are all real. Then solve for $a,b,c$ and $d$, there must be a set of real solution.