I am trying to find the map:
$$\psi_2(u,w,v):\mathbb{S}^2\smallsetminus\{0,0,-1\}\rightarrow\mathbb{C}$$
So this is what I have done, let $(u,v,w)$ be a point on the sphere, and let $(0,0,-1)$ be the south pole. The line between these two points can be given by:
$$\mathbf{r}(t)=(u,v,w)t+(1-t)(0,0,-1)$$ $$=(tu,tv,tw-1+t)$$ Now if I want to find the point on the plane $z=0$ I have to have: $$tw-1+t=0$$ $$\Rightarrow t=\frac{1}{1+w}$$ Which then gives: $$(x,y)=(\frac{u}{1+w},\frac{v}{1+w})$$ Which can be associated with the complex number: $$z=\frac{u}{1+w}+i\frac{v}{1+w}$$ However wikipedia says that his map should be: $$\psi(u,v,w)=\frac{u}{1+w}-i\frac{v}{1+w}$$ And I can't figure out where I went wrong. When I did the same thing for the northpole projection I got exactly the result that wikipedia got. So where am I going wrong?
You haven't gone wrong, calculation-wise. The problem, as you've identified, is that the two stereographic projection mappings from the north and south poles are not holomorphically compatible; their composition is inversion in the unit circle, $$ z \mapsto \frac{z}{|z|^{2}} = \frac{1}{\bar{z}}, $$ rather than the desired complex reciprocal map. If we take stereographic projection from the north pole as one local holomorphic chart, then the complex conjugate of stereographic projection from the south pole is holomorphically compatible, however.