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Plot modular function on unit disk

I have been using the following algorithm to plot some complex functions in Mathematica. I am trying to get a representation of modular function on the unit disk using Mathematica like in this link (not produced by mathematica) 
I have tried modifying the function below using Cayley transform with no success. 

I would very much appreciate any suggestions. 



ComplexGraph[f_, xmin_, xmax_, ymin_, ymax_, points_: 100] := 
 (* f is the complex function to be graphed in the region
 [xmin,xmax] * [ymin,ymax]. 
 The parameter points controls how many points will be
  sampled in each direction;. good values are 100-500. *)
 RegionPlot[True, {x, xmin, xmax}, {y, ymin, ymax},
  ColorFunction -> Function[{x, y},
    Hue[Mod[Arg[f[(I - (x + I*y))/(I + (x + I*y))]], 2 Pi]/(2 Pi),
     1/(1 + 0.3 Log[Abs[f[(I - (x + I*y))/(I + (x + I*y))]] + 1]),
     1 - 1/(1.1 + 
         5 Log[Abs[f[(I - (x + I*y))/(I + (x + I*y))]] + 1])]],
  ColorFunctionScaling -> False,
  AspectRatio -> Automatic,
  PlotPoints -> points]

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