       Re: Plot modular function on unit disk

• To: mathgroup at smc.vnet.net
• Subject: [mg109579] Re: Plot modular function on unit disk
• From: Patrick Scheibe <pscheibe at trm.uni-leipzig.de>
• Date: Thu, 6 May 2010 04:51:43 -0400 (EDT)

```Hi,

I would use RegionPlot. Why don't you create your raster of pixels with
(Parallel)Table und use ArrayPlot to create the graphics?
ParallelTable is
fast and you can create rasters of arbitrary size.

nx=512;
cmplxFunc[x_, y_] :=
If[Abs[x + I y] != 0,
With[{nome = 1/(I Pi) Log[x + I y]},
If[Im[nome] > 0.009,
With[{value = KleinInvariantJ[nome]},
Min[2^15, Max[0, Re[value]]]
], 0]
], 0]
DistributeDefinitions[cmplxFunc,nx]

data = ParallelTable[
cmplxFunc[x, y], {y, -1, 1, 2./nx}, {x, -1, 1, 2./nx}];

ArrayPlot[data,
ColorFunction -> (If[# == 0., Lighter[Gray],
ColorData["GreenPinkTones"][#^0.2]] &)]

Please note that you can use triples in ArrayPlot too if you want to set
angle, saturation and brightness of hue:

ArrayPlot[RandomReal[{0, 1}, {3, 3, 3}], ColorFunction -> (Hue[##] &)]

Cheers
Patrick

Am May 3, 2010 um 1:55 PM schrieb Snas:

> 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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