I am studying Hyperbolic Geometry on my own. More especially, I am studying the Moebius transformation of the upper half plane model in hyperbolic geometry. But I got stuck studying it. I am looking for a solution of this problem. The problem is as follows:
Find the Moebius transformation that rotates the hyperbolic plane about $i$ through an angle of $\frac { \pi}{4}$.
Please help me. Thanking in advanced.
The stabilizer of $i$ in the upper half-plane consists of elliptic linear-fractional transformations $g_A(z)=\frac{az+b}{cz+d}$ corresponding to matrices $$ A=\left[\begin{array}{cc} a & b\\ c & d \end{array}\right]= \left[\begin{array}{cc} \cos(\phi) & \sin(\phi)\\ -\sin(\phi) & \cos(\phi) \end{array}\right] $$ (which makes it easy to remember: Euclidean rotations correspond to hyperbolic rotations). The angle of rotation $\phi$ of the matrix $A$ acting on ${\mathbb R}^2$ correspond to the angle of rotation $2\phi$ for the Moebius transformation $g_A$. (This is again easy to remember since for $A=-I$, rotation by $\pi$, $g_A=id_{\mathbb H^2}$, rotation by $2\pi$.) The rest you can figure out on your own.