In[]:=
myLog[z_,θ_:0]:=Log[Abs[z]]+I(Arg[zExp[Iθ]]-θ);​​myNthRoot[z_,n_Integer/;n≠0,θ_:0]:=Exp[myLog[z,θ]/n];

ArcCos & ArcSin

In[]:=
ComplexPlot3D[​​π/2+Log[z+myNthRoot[1-z,2,-π]myNthRoot[1+z,2]]-myLog[z,π/2],​​{z,4},PlotRangeFull]
Out[]=
In[]:=
ComplexPlot3D[​​Log[z+myNthRoot[1-z,2,-π]myNthRoot[1+z,2]]-myLog[z,π/2],​​{z,4},PlotRangeFull]
Out[]=

Sqrt

In[]:=
ComplexPlot3D[​​myNthRoot[1-z,2]myNthRoot[1+z,2],​​{z,4},PlotRangeFull]
Out[]=
In[]:=
ComplexPlot3D[ myNthRoot[1 - z, 2] myNthRoot[1 + z, 2, -π], {z, 4}, PlotRange  Full]
Out[]=

ArcTan

In[]:=
ComplexPlot3D[​​-/2myLog[(-z)/(+z)],​​{z,4},PlotRangeFull]
Out[]=
In[]:=
ComplexPlot3D[​​-/2(myLog[-z,π/2]-myLog[+z,-π/2]),​​{z,4},PlotRangeFull]
Out[]=