Plot modular function on unit disk

*To*: mathgroup at smc.vnet.net*Subject*: [mg109532] Plot modular function on unit disk*From*: Snas <naso at netspace.net.au>*Date*: Mon, 3 May 2010 07:55:35 -0400 (EDT)

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) http://en.wikipedia.org/wiki/File:J-inv-real.jpeg I have tried modifying the function below using Cayley transform with no success. I would very much appreciate any suggestions. Sincerely, Snaes 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]