So for my Complex Analysis class, I need to prove the following question:
Consider the function that maps a point from $\mathbb{C} \cup \{\infty\}$ to the sphere via inverse stereographic projection, applies a rotation to the sphere and then maps it back to the complex plane by inverse stereographic projection. Show that this function is a Möbius transformation.
Now, we use the complex analysis book of Stein and Shakarchi, where this sphere has center $(0,0,1/2)$ and radius $1/2$. The stereographic projection is defined as
$x = \frac{X}{1-Z}$ and $y = \frac{Y}{1-Z}$
and its inverse as
$X = \frac{x}{x^2+y^2+1}$, $Y = \frac{y}{x^2+y^2+1}$ and $Z = \frac{x^2+y^2}{x^2+y^2+1}$.
I managed to prove that this is indeed a Möbius transformation when the $Z$-axis is the rotation axis but I have no clue how I could prove this for every rotation of the sphere.