ϕ[u_]:=u-ArcTan[u];
a[u_,v_]:=(2/(1+(u^2)((Sin[v])^2)));
r[u_,v_]:=a[u,v](Sqrt[1+(u^2)])Sin[v];
x[u_,v_]:=r[u,v]Cos[ϕ[u]]
y[u_,v_]:=r[u,v]Sin[ϕ[u]]
z[u_,v_]:=Log[Tan[(v/2)]]+a[u,v]Cos[v]

ParametricPlot3D[{x[u,v],y[u,v],z[u,v]}, {u,0,2π},{v,0,π}, PlotPoints -> {100,100}, ViewPoint -> 10{5,1,0}]

Clear[x,y,z];

x[u_,v_]:=((2(Cos[u]+u Sin[u])Sin[v])/(1+(u^2)((Sin[v])^2)))
y[u_,v_]:=((2(Sin[u]-u Cos[u])Sin[v])/(1+(u^2)((Sin[v])^2)))
z[u_,v_]:=Log[Tan[(v/2)]]+((2Cos[v])/(1+(u^2)((Sin[v])^2)))

ParametricPlot3D[{x[u,v],y[u,v],z[u,v]}, {u,-4,4},{v,0,π-0.01}, PlotPoints -> {50,100}, ViewPoint -> 10{5,1,0}, Boxed->False,Axes->False,Mesh->False, PlotRange ->1.2{-2,2}, ImageSize->200]

ParametricPlot3D[{x[u,v],y[u,v],z[u,v]}, {u,0,2π+π/2-0.01},{v,0,π-0.01}, PlotPoints -> {50,100}, ViewPoint -> 10{5,1,0}, Boxed->False,Axes->False,Mesh->False, PlotRange ->1.2{-2,2}, ImageSize->200]