Find the hyperbolic distance between $2$ and $5+i$ in the upper half plane $H=\{ z: Im(z)>0\}$.

Question

Find the hyperbolic distance between $2$ and $5+i$ in the upper half plane $H=\{ z: Im(z)>0\}$.

Ans: we know the metric $d_H(z, w)=2\tanh^{-1}(|\frac{z-w}{z-\bar w}|)$ then $d_H(2, 5+i)=2\tanh^{-1}(|\frac{-3-i}{-3+i}|)=2\tanh^{-1}(1)$ is tend to $\infty$ as $x=1$ is asymptotic to $\tanh^{-1}x$. Is it correct? If wrong please correct me.

Answer 1

Your calculation is correct. However, $2\not \in H$.
If you think about the line $Im(z)=0$ as points/the point at infinity, then the result sill makes sense.