WE[f_,g_,zeta_]:=ComplexExpand[Re[Integrate[phi[f,g,z,#],{z,0,zeta},GenerateConditions->False]]&/@Range[3]]
phi[f_,g_,z_,1]:=f[z](1-g[z]^2)/2
phi[f_,g_,z_,2]:=I f[z](1+g[z]^2)/2
phi[f_,g_,z_,3]:=f[z]g[z]

Clear[wr]
wr[A_]=WE[#&,#^-2&,Exp[R] Exp[I phi]]//FullSimplify

Manipulate[
ParametricPlot3D[
wr[A]//Evaluate,
{R,-.5,Log[.5]},{phi,0,Pi},
(*{u,.0,3},{v,.1,3},*)
BoundaryStyle->Directive[Thick,Red],
PlotRange->All,
BoxRatios->{1,1,1},
Boxed->False,
Axes->False,
MaxRecursion->4,
MeshFunctions->{#4&},
Mesh->14,
MeshShading->ControlActive[None,{Pink,White}],
PlotStyle->None,
PlotPoints->{30,30},
Mesh->False,
RotationAction->"Clip",
PlotRange->{{-1,1},{-1,1},{-1,1}}
],
{A,0,1}]