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]

```

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