I have the following function that is for the stereographic projection: $$f(x, y) = \dfrac{1}{\kappa}(2x, 2y, x^2 + y^2 - 1)\hspace{0.3cm} \text{where} \hspace{0.3cm} \kappa = x^2 + y^2 + 1.$$ So let now I calculate the partial derivatives of the function, giving me this $$Dfe_{1} = \dfrac{\partial f}{\partial x} = \dfrac{2}{\kappa^2}(- x^2 + y^2 + 1, -2xy, 2x)$$ and $$Dfe_{1} = \dfrac{\partial f}{\partial y} = \dfrac{2}{\kappa^2}(-2xy, x^2 - y^2 + 1, 2y)$$ where $e_{i}$ is a canonical basis which implies that $f$ is conformal.
I am trying to prove that the stereographic projection is antiholomorphic, due to the fact that by definition, the function is anticonformal (preserves magnitud of angles but changes the orientation). However, any holomorphic or antiholomorphic function is conformal, so I am trying to find the anticonformal property. Does anyone know how to demonstrate that the stereogrpahic projection is antiholomorphic but algebraically ?
The main reason I am trying to prove this is that I could use the Cauchy-Riemann equations to show the sign of the Jacobean is strictly negative.